ALGEBRAIC VARIETIES WITH MANY RATIONAL POINTS Contentstschinke/papers/yuri/08cmi/cmi1.pdf · of...

ALGEBRAIC VARIETIES WITH MANY RATIONALPOINTS

YURI TSCHINKEL

Contents

Introduction 11. Geometry background 32. Existence of points 203. Density of points 284. Counting problems 355. Counting points via universal torsors 636. Height zeta functions 71References 89

Introductionsect:introduction

Let f ∈ Z[t, x1, . . . , xn] be a polynomial with coefficients in the inte-gers. Consider

f(t, x1, . . . , xn) = 0,

as an equation in the unknowns t, x1, . . . , xn or as an algebraic family ofequations in x1, . . . xn parametrized by t. We are interested in integersolutions: in their existence and distribution. Sometimes the emphasisis on individual equations, e.g.,

xn + yn = zn,

sometimes we want to understand a typical equation, i.e., a generalequation in some family. To draw inspiration (and techniques) fromdifferent branches of algebra it is necessary to consider solutions withvalues in other rings and fields, most importantly, finite fields Fq, finiteextensions of Q, or the function fields Fp(t) and C(t). While thereis a wealth of ad hoc elementary approaches to individual equations,

Date: July 21, 2008.The second author was supported by the NSF grant 0602333.

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and deep theories focussing on their visible or hidden symmetries, ourprimary approach here will be via geometry.

Basic geometric objects are the affine space An and the projectivespace Pn = (An+1 \ 0) /Gm, the quotient by the diagonal action of themultiplicative group. Concretely, affine algebraic varieties Xaffine ⊂ An

are defined by systems of polynomial equations with coefficients insome base ring R; their solutions with values in R, Xaffine(R), arecalled R-integral points. Projective varieties are defined by homoge-neous equations, and their R-points are equivalence classes of solutions,with respect to diagonal multiplication by nonzero elements in R. IfF is the fraction field of R, then Xprojective(R) = Xprojective(F ), andthese points are called F -rational points. The geometric advantagesof working with “compact” projective varieties translate to importanttechnical advantages in the study of equations, and the theory of ra-tional points is currently much better developed.

The sets X(F ) reflect on the one hand the geometric and algebraiccomplexity of X (e.g., the dimension of X), and on the other handthe structure of the ground field F (e.g., its topology, analytic struc-ture). It is important to consider the variation of X(F ′), as F ′ runsover extensions of F , either algebraic extensions, or completions. It isalso important to study projective and birational invariants of X, itsbirational models, automorphisms, fibration structures, deformations.Each point of view contributes its own set of techniques, and it is theinteraction of ideas from a diverse set of mathematical cultures thatmakes the subject so appealing and vibrant.

The focus in these notes will be on smooth projective varieties Xdefined over Q, with many Q-rational points. Main examples are vari-eties Q-birational to Pn and hypersurfaces in Pn of low degree. We willstudy the relationship between the global geometry of X over C andthe distribution of rational points in Zariski topology and with respectto heights. Here are the problems we face:

• Existence of solutions: local obstructions, the Hasse principle,global obstructions;• Density in various topologies: Zariski density, weak approxima-

tion;• Distribution with respect to heights: bounds on smallest points,

asymptotics.

VARIETIES WITH MANY RATIONAL POINTS 3

Here is the roadmap of the paper. Section 1 contains a summaryof basic terms from complex algebraic geometry: main invariants ofalgebraic varieties, classification schemes, and examples most relevantto arithmetic in dimension ≥ 2. Section 2 is devoted to the existenceof rational and integral points, including aspects of decidability, effec-tivity, local and global obstructions. In Section 3 we discuss Lang’sconjecture and its converse, focussing on varieties with nontrivial en-domorphisms and fibration structures. Section 4 introduces heights,counting functions, and height zeta functions. We explain conjecturesof Batyrev, Manin, Peyre and their refinements. The remaining sec-tions are devoted to geometric and analytic techniques employed inthe proof of these conjectures: universal torsors, harmonic analysis onadelic groups, p-adic integration and “estimates”.

Acknowledgments. I am very grateful to V. Batyrev, F. Bogomolov,U. Derenthal, A. Chambert-Loir, J. Franke, J. Harris, B. Hassett, A.Kresch, Y. Manin, E. Peyre, J. Shalika, M. Strauch and R. Takloo-Bighash for the many hours of listening and sharing their ideas. Partialsupport was provided by National Science Foundation Grants 0554280and 0602333.

1. Geometry backgroundsect:geometry

We discuss basic notions and techniques of algebraic geometry thatare commonly encountered by number theorists. For most of this sec-tion, F is an algebraically closed field of characteristic zero. Geometryover algebraically closed fields of positive characteristic, e.g., algebraicclosure of a finite field, differs in several aspects: difficulties arisingfrom inseparable morphisms, “unexpected” maps between algebraicvarieties, additional symmetries, lack (at present) of resolution of sin-gularities. Geometry over nonclosed fields, especially number fields,introduces new phenomena: varieties may have forms, not all geomet-ric constructions descend to the ground field, parameter counts do notsuffice. In practice, it is “equivariant geometry for finite groups”, withGalois-symmetries acting on all geometric invariants and special loci.The case of surfaces is addressed in detail in [Has08].

sect:pic1.1. Basic invariants. Let X be an algebraic variety over F . We mayassume that X is projective and smooth. We seek to isolate invariantsof X that are most relevant for arithmetic investigations.

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There are two natural types of invariants: birational invariants, i.e.,invariants of the function field F (X), and projective geometry invari-ants, i.e., those arising from a concrete representation of X as a sub-variety of Pn. Examples are the dimension dim(X), defined as thetranscendence degree F (X) over F , and the degree of X in the givenprojective embedding. For hypersurfaces Xf ⊂ Pn the degree is simplythe degree of the defining homogeneous polynomial. In general, it isdefined via the Hilbert function of the homogeneous ideal, or geomet-rically, as the number of intersection points with a general hyperplaneof codimension dim(X).

The degree alone is not a sensitive indicator of the complexity ofthe variety: Veronese embeddings of P1 → Pn exhibit it as a curve ofdegree n. In general, we may want to consider all possible projectiveembeddings of a variety X. Two such embeddings can be “composed”via the Segre embedding Pn × Pm → PN , where N = nm + n + m.For example, we have the standard embedding P1 × P1 → P3, withimage a smooth quadric. In this way, projective embeddings of Xform a “monoid”; the corresponding abelian group is the Picard groupPic(X). Alternatively, it is the group of isomorphism classes of linebundles on X. Cohomologically,

Pic(X) = H1et(X,Gm),

where Gm is the sheaf of invertible functions. Yet another descriptionis

Pic(X) = Div(X)/ (C(X)∗/C∗) ,where Div(X) is the free abelian group generated by codimension onesubvarieties of X, and C(X)∗ is the multiplicative group of rationalfunctions of X, each f ∈ C(X)∗ giving rise to a principal divisor div(f)(divisor of zeroes and poles of f). Sometimes it is convenient to identifydivisors with their classes in Pic(X). Note that Pic is a contraviariantfunctor: a morphism X → X induces a homomorphism of abeliangroups Pic(X)→ Pic(X). There is an exact sequence

1→ Pic0(X)→ Pic(X)→ NS(X)→ 1

where Pic0(X) is the connected component of the identity in Pic(X)and NS(X) is the Neron-Severi group of X. In most applications inthis paper, Pic0(X) is trivial.

Given a projective variety X ⊂ Pn, via an explicit system of homo-geneous equations, we can easily write down at least one divisor onX, a hyperplane section L in this embedding. Another divisor, the


divisor of of zeroes of a differential form of top degree on X, can alsobe computed from the equations. Its class KX ∈ Pic(X), i.e., the class

of the line bundle Ωdim(X)X , is called the canonical class. In general, it

is not known how to write down effectively divisors whose classes arenot proportional to linear combinations of KX and L. This can bedone for some varieties over Q, e.g., smooth cubic surfaces in X3 ⊂ P3

(see Section 1.9), but is already open for smooth quartics X4 ⊂ P4 (forsome partial results in this direction, see Section 1.10).

Elements in Pic(X) corresponding to projective embeddings generatethe ample cone Λample(X) ⊂ Pic(X)R; ample divisors arise as hyper-plane sections ofX in a projective embedding. The closure of Λample(X)in Pic(X)R is called the nef cone. An effective divisor is a sum withnonnegative coefficients of irreducible subvarieties of codimension one.Their classes span the effective cone Λeff(X). Divisors giving rise toembeddings of some Zariski open subset of X form the big cone. Tosummarize we have

Λample(X) ⊆ Λnef(X) ⊆ Λbig(X) ⊆ Λeff(X) ⊂ Pic(X)R.

These cones and their combinatorial structure encode important geo-metric information. For example, for all divisors D ∈ Λnef(X) and allcurves C ⊂ X, the intersection number D.C ≥ 0 [Kle66]. Divisors onthe boundary of Λample(X) give rise to fibration structures on X; wewill discuss this in more detail in Section 1.4.

exam:blowup Example 1.1.1. Let X be a smooth projective variety, Y ⊂ X a smoothsubvariety and π : X = BlY (X) → X the blowup of X in Y , i.e., thecomplement in X to the exceptional divisor E := π−1(Y ) is isomorphicto X \ Y , and E itself can be identified with the projectivized tangentcone to X at Y . Then

Pic(X) ' Pic(X)⊕ ZE

and

KX = π∗(KX) +O((codim(Y )− 1)E)

(see [Har77, Exercise 8.5]). Note that

π∗(Λeff(X)) ⊂ Λeff(X),

but that pullbacks of ample divisors are not necessarily ample.

exam:hypersurface Example 1.1.2. Let X ⊂ Pn be a hypersurface of dimension ≥ 3 anddegree d. Then Pic(X) = NS(X) = ZL, generated by the class of the

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hyperplane section, and

Λample(X) = Λeff(X) = R≥0L.

The canonical class is

KX = −(n+ 1− d)L.

exam:cubic-cone Example 1.1.3. If X is a smooth cubic surface over an algebraicallyclosed field, then Pic(X) = Z7. The anticanonical class is proportionalto the sum of 27 exceptional curves (lines):

−KX =1

9(D1 + · · ·+D27).

The effective cone Λeff(X) ⊂ Pic(X)R is spanned by the classes of thelines.

On the other hand, the effective cone of a minimal resolution of thesingular cubic surface

x0x23 + x2

1x3 + x32 = 0

is a simplicial cone (in R7) [HT04].

exam:solvable Example 1.1.4. Let G be a connected solvable linear algebraic group,e.g., the additive group G = Ga, an algebraic torus G = Gd

m or thegroup of upper-triangular matrices. Let X be an equivariant compact-ification of G, i.e., the action of G on itself by extends to X. Usingequivariant resolution of singularieties, if necessary, we may assumethat X is smooth projective and that the boundary

X \G = D = ∪i∈I Di, with Di irreducible,

is a divisor with normal crossings. Every divisor D on X is equivalentto a divisor with support in the boundary since it can be “moved” thereby the action of G. Thus Pic(X) is generated by the components Di,and the relations are given by functions with zeroes and poles supportedin D, i.e., by the characters X∗(G). We have an exact sequence

eqn:exact-coneeqn:exact-cone (1.1) 0→ X∗(G)→ ⊕i∈I ZDiπ−→ Pic(X)→ 0

The cone of effective divisors Λeff(X) ⊂ Pic(X)R is the image of thesimplicial cone ⊕i∈I R≥0Di under the projection π. The anticanonicalclass is

−KX = ⊕i∈I κiDi, with κi ≥ 1, for all i.

For unipotent G one has κi ≥ 2, for all i [HT99].


For higher-dimensional varieties without extra symmetries, the com-putation of the ample and effective cones, and of the position of KX

with respect to these cones, is a difficult problem. A sample of recentpapers on this subject is: [CS06], [Far06], [FG03], [Cas07], [HT03],[HT02], [GKM02]. However, we have the following fundamental result(see also Section 1.4):

thm:fg-cones Theorem 1.1.5. Let X be a smooth projective variety with −KX ∈Λample(X). Then Λnef(X) is a finitely generated rational cone. If −KX

is big and nef then Λeff(X) is finitely generated.

Finite generation of the nef cone goes back to Mori (see [CKM88]for an introduction). The result concerning Λeff(X) has been proved in[Bat92] in dimension ≤ 3, and in higher dimensions in [BCHM06] (seealso [Ara05], [Leh08]).

sect:schemes1.2. Classification schemes. In some arithmetic investigations (e.g.,Zariski density or rational points) we rely mostly on birational proper-ties of X; in others (e.g., asymptotics of points of bounded height), weneed to work in a fixed projective embedding.

Among birational invariants, the most important are those arisingfrom a comparison of X with a projective space:

(1) rationality: there exists a birational isomorphism X ∼ Pn, i.e.,the is an isomorphisms of function fields F (X) = F (Pn), forsome n ∈ N;

(2) unirationality: there exists a dominant map Pn 99K X;(3) rational connectedness: for general x1, x2 ∈ X(F ) there exists

a morphism f : P1 → X such that x1, x2 ∈ f(P1).

It is easy to see that(1)⇒ (2)⇒ (3).

These properties are equivalent in dimension two, but diverge in higherdimensions. First examples of unirational but not rational threefoldswere constructed in [IM71] and [CG72]. The approach of [IM71] was tostudy of the group Bir(X) of birational automorphisms of X; finitenessof Bir(X), i.e., birational rigidity, implies nonrationality. No examplesof smooth projective rationally connected but not unirational varietiesare known.

Interesting unirational varieties arise as quotients V/G, where V =An is a representation space for a faithful action of a linear algebraicgroup G. For example, the moduli spaceM0,n of n points on P1 is bira-tional to (P1)n/PGL2. Moduli spaces of degree d hypersurfaces X ⊂ Pn

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are naturally isomorphic to P(Symd(An+1))/PGLn+1. Rationality ofV/G is known as Noether’s problem. It has a positive solution for Gbeing the symmetric group Sn, the group PGL2 [Kat82], [BK85], andin many other cases [SB89], [SB88]. Counterexamples for some finiteG were constructed in [Sal84], [Bog87]; nonrationality is detected bythe unramified Brauer group, Brun(V/G), closely related to the Brauergroup of the function field Br(F (V/G)) = H2

et(F (V/G),Gm).Now we turn to invariants arising from projective geometry, i.e., am-

ple line bundles on X. For smooth curves C, an important invariant isthe genus g(C) := dim H0(X,KX). In higher dimensions, one considersthe Kodaira dimension

eqn:kodaira-dimeqn:kodaira-dim (1.2) κ(X) := lim suplog(dim H0(X,nKX))

log(n),

and the related graded canonical section ring

eqn:graded-kxeqn:graded-kx (1.3) R(X,KX) = ⊕n≥0 H0(X,nKX).

A fundamental theorem is that this ring is finitely generated [BCHM06].The Kodaira dimension is the dimension of the variety Proj(R(X,KX)),or equivalently, the dimension of the image of X under the map

X → P(H0(X,nKX)),

for sufficiently large n. For KX ample one has κ(X) = dim(X).A very rough classification of smooth algebraic varieties is based

on the position of the anticanonical class with respect to the cone ofample divisors. Numerically, this is reflected in the value of the Kodairadimension. There are three main cases:

• Fano: −KX ample;• general type: KX ample;• intermediate type: none of the above.

For curves, this classification can be read off from the genus: curves ofgenus 0 are of Fano type, of genus 1 of intermediate type, and of genus≥ 2 of general type. Other examples of varieties in each group are:

• Fano: Pn, smooth degree d hypersurfaces Xd ⊂ Pn, with d ≤ n;• general type: hypersurfaces Xd ⊂ Pn, with d ≥ n + 2, moduli

spaces of curves of high genus and abelian varieties of highdimension;• intermediate type: P2 blown up in 9 points, abelian varieties,

Calabi-Yau varieties.


There are only finitely many families of smooth Fano varieties ineach dimension [KMM92]. On the other hand, the universe of varietiesof general type is boundless and there are many open classificationquestions already in dimension 2. The qualitative behavior of rationalpoints on X is closely related to this classification (see Section 3). Inour arithmetic applications we will mostly encounter Fano varieties andvarieties of intermediate type.

In finer classification schemes such as the Minimal Model Program(MMP) it is important to take into account fibration structures andmild singularities (see [KMM87] and [Cam04]). Indeed, recall thatR(X,KX) is finitely generated and put Y = Proj(R(X,KX)). Thenthe general fiber of the rational projection

X 99K Y = Proj(R(X,KX)),

is a (possibly singular) Fano variety. For example, a surface of Kodairadimension 1 is birational to a Fano fiber space over a curve of genus≥ 1.

Analogously, in many arithmetic questions, the passage to fibrationsis inevitable (see Section 4.16). These often arise from the section rings

eqn:rxleqn:rxl (1.4) R(X,L) = ⊕n≥0 H0(X,nL).

Consequently, one considers the Iitaka dimension

eqn:iitaka-dimeqn:iitaka-dim (1.5) κ(X,L) := lim suplog(dim H0(X,nL))

log(n).

Finally, a pair (X,D), where X is smooth projective and D is a divisorin X, gives rise to another set of invariants: the log Kodaira dimen-sion κ(X,KX + D) and the log canonical ring R(X,KX + D), whosefinite generation is also known in many cases [BCHM06]. Again, onedistinguishes

• log Fano: κ(X,−(KX +D)) = dim(X);• log general type: κ(X,KX +D) = dim(X);• log intermediate type: none of the above.

This classification has consequences for the study of integral points onthe open variety X \D.

sect:sing1.3. Singularities. Assume that X is Q-Cartier, i.e., there exists aninteger m such that mKX is a Cartier divisor. Let X be a normalvariety and f : X → X a proper birational morphism. Denote by E

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the f -exceptional divisor and by e its generic point. Let g = 0 be alocal equation of E. Locally, we can write

f ∗(generator of O(mKX)) = gmd(E)(dy1 ∧ . . . ∧ dyn)m

for some d(E) ∈ Q such that md(E) ∈ Z. If, in addition, KX is a line

bundle (e.g., X is smooth), then mKX is linearly equivalent to

f ∗(mKX) +∑i

m · d(Ei)Ei; Ei exceptional,

and numerically

KX ∼ f ∗(KX) +∑i

d(Ei)Ei.

The number d(E) is called the discrepancy of X at the exceptionaldivisor E. The discrepancy discr(X) of X is

discr(X) := infd(E) | all f, EIf X is smooth then discr(X) = 1. In general, discr(X) ∈ −∞ ∪[−1, 1].

defin:singularities Definition 1.3.1. The singularities of X are called

• terminal if discr(X) > 0 and• canonical if discr(X) ≥ 0.

It is essential to remember that terminal = smooth in codimension2 and that for surfaces, canonical means Du Val singularities.

Canonical isolated singularities on surfaces can be classified via Dynkindiagrams: Let f : X → X be the minimal desingularization. Thenthe submodule in Pic(X) spanned by the classes of exceptional curves(with the restriction of the intersection form) is isomorphic to the rootlattice of the corresponding Dynkin diagram (exceptional curves givesimple roots).

Canonical singularities don’t influence the expected asymptotic forrational points on the complement to all exceptional curves: for (sin-gular) Del Pezzo surfaces X we still expect an asymptotic of points ofbounded anticanonical height of the shape B log(B)9−d, where d is thedegree of X, just like in the smooth case (see Section 4.12). This failswhen the singularities are worse than canonical.

exam:weight Example 1.3.2. Let w = (w0, . . . , wn) ∈ Nn, with gcd(w0, . . . , wn) = 1and let

X = X(w) = P(w0, . . . , wn)


be a weighted projective space, i.e., we have a quotient map

(An+1 \ 0)Gm−→ X,

where the torus Gm acts by

λ · (x0, . . . , xn+1) 7→ (λw0x0, . . . , λwnxn).

For w = (1, . . . , 1) it is the usual projective space, e.g., P2 = P(1, 1, 1).The weighted projective plane P(1, 1, 2) has a canonical singularity andthe singularity of P(1, 1,m), with m ≥ 3, is worse than canonical.

For a discussion of singularieties on general weighted projective spacesand so called fake weighted projective spaces see, e.g., [Kas08].

sect:mmp1.4. Minimal Model Program. Here we recall basic notions fromthe Minimal Model Program (MMP) (see [CKM88], [KM98], [KMM87],[Mat02] for more details). The starting point is the following funda-mental theorem due to Mori [Mor82]:

thm:mori Theorem 1.4.1. Let X be a smooth Fano variety of dimension n.Then there exists an integer d ≤ n + 1 such that through every geo-metric point x of X there passes a rational curve of −KX-degree ≤ d.

These rational curves move in families. Their specializations arerational curves, which may move again, and again, until one arrives at“rigid” rational curves.

Theorem 1.4.2 (Cone theorem). Let X be a smooth Fano variety.Then the closure of the cone of (equivalence classes of) effective curvesin H2(X,R) is finitely generated by classes of rational curves.

The generating rational curves are called extremal rays, they corre-spond to codimension-1 faces of the dual cone of nef divisors. Mori’sMinimal Model Program links the convex geometry of the nef coneΛnef(X) with natural transformations of X. Pick a divisor D on theface dual to an extremal ray [C]. It is not ample anymore, but it stilldefines a map

X → Proj(R(X,D)),

which contracts the curve C to a point. The map is one of the following:

• a fibration over a base of smaller dimension, and the restrictionof D to a general fiber proportional to the anticanonical classof the fiber, which is a (possibly singular) Fano variety,• a birational map contracting a divisor,• a contraction of a subvariety in codimension ≥ 2 (a small con-

traction).

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The image could be singular, as in Example 1.3.2, and one of the mostdifficult issues of MMP was to develop a framework which allows tomaneuver between birational models with singularities in a restrictedclass, while keeping track of the modifications of the Mori cone ofcurves. In arithmetic applications, for example proofs of the existenceof rational points as in, e.g., [CTSSD87a], [CTSSD87b], [CTS89], onerelies on the fibration method and descent, applied to some auxiliaryvarieties. Finding the “right” fibration is an art. Mori’s theory gives asystematic approach to these questions.

A variant of Mori’s theory, the Fujita program, analyzes fibrationsarising from divisors on the boundary of the effective cone Λeff(X). Inthis case, the restriction of D to a general fiber is a perturbation of(some positive rational multiple of) the anticanonical class of the fiberby a rigid effective divisor. This theory turns up in the analysis ofheight zeta functions in Section 6 (see also Section 4.15).

We will need the notion of the geometric hypersurface of linear growth:

eqn:lin-geomeqn:lin-geom (1.6) ΣgeomX := L ∈ NS(X)R | a(L) = 1

Let X be smooth projective with Pic(X) = NS(X) and a finitelygenerated effective cone Λeff(X). For a line bundle L on X define

eqn:a(L)eqn:a(L) (1.7) a(L) := min(a | aL+KX ∈ Λeff(X)).

Let b(L) be the maximal codimension of the face of Λeff(X) containinga(L)L+KX . In particular,

a(−KX) = 1 and b(−KX) = rk Pic(X).

These invariants arise in Manin’s conjecture in Section 4.12 and theanalysis of analytic properties of height zeta functions in Section 6.1.

sect:campana1.5. Campana’s program. Recently, Campana developed a new ap-proach to classification of algebraic varieties with the goal of formulat-ing necessary and sufficient conditions for potential density of rationalpoints. The key notions are: the core of an algebraic variety and specialvarieties. Special varieties include Fano varieties and Calabi–Yau vari-eties. They are conjectured to have a potentially dense set of rationalpoints. This program is explained in detail in [Abr08].

sect:cox1.6. Cox rings. Again, we assume that X is a smooth projective va-riety with Pic(X) = NS(X). Examples are Fano varieties, equivariantcompactifications of algebraic groups and holomorphic symplectic va-rieties. Fix line bundles L1, . . . , Lr whose classes generate Pic(X). For


n = (n1, . . . , nr) ∈ Zr we put

Ln := Ln11 ⊗ . . .⊗ Lnrr .

The Cox ring is the multigraded section ring

Cox(X) := ⊕n∈ZrH0(X,Ln).

The key issue is finite generation of this ring. This has been proved un-der quite general assumptions in [BCHM06, Corollary 1.1.9]. Assumethat Cox(X) is finitely generated. Then both Λeff(X) and Λnef(X)are finitely generated polyhedral (see [HK00, Proposition 2.9]). Otherimportant facts are:

(1) X is a toric variety if and only if Cox(X) is a polynomial ring[Cox95], [HK00, Corollary 2.10].

(2) Cox(X) is multigraded for NS(X), in particular, it carries anatural action of the dual torus TNS.

sect:cox1.7. Universal torsors. We continue to work over an algebraicallyclosed field. Let G be a linear algebraic group and X an algebraicvariety. A G-torsor over X is a principal G-bundle π : TX → X. Ba-sic examples are GLn-torsors, they arise from vector bundles over X;for instance, each line bundle L gives rise to a GL1 = Gm-torsor overX. Up to isomorphism, G-torsors are classified by H1

et(X,G); line bun-dles are classified by H1

et(X,Gm) = Pic(X). When G is commutative,H1et(X,G) is a group.Let G = Gd

m be an algebraic torus and X∗(G) = Zd its characterlattice. A G-torsor over an algebraic variety X is determined, up toisomorphism, by a homomorphism

χ : X∗(G)→ Pic(X).

Assume that Pic(X) = NS(X) = Zd and that χ is in fact an isomor-phism. The arising G-torsors are called universal. The introduction ofuniversal torsors is motivated by the fact that over nonclosed fields they“untwist” the action of the Galois group on the Picard group of X (seeSections 1.13 and 2.5). The “extra dimensions” and “extra symme-tries” provided by the torsor add crucial freedom in the analysis of thegeometry and arithmetic of the underlying variety. Examples of appli-cations to rational points will be presented in Sections 2.5 and 5. Thisexplains the surge of interest in explicit equations for universal torsors,the study of their geometry: singularities and fibration structures.

Assume that Cox(X) is finitely generated. Then Spec (Cox(X)) con-tains a universal torsor TX of X as an open subset.

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[Cox95][STV06], [TVAV08], [SS07], [SS08], book [Sko01][HK00]

sect:hyper1.8. Hypersurfaces. We now turn from the general theory to specificvarieties. Let X ⊂ Pn be a smooth hypersurface of degree d. We havealready described some of its invariants in Example 1.1.2, at least whendim(X) ≥ 3. In particular, in this case Pic(X) ' Z. In dimension two,there are more possibilities.

The most interesting cases are d = 2, 3, and 4. A quadric X2 isisomorpic to P1 × P1 and has Picard group Pic(X2) ' Z⊕ Z. A cubichas Picard group of rank 7. These are examples of Del Pezzo surfacesdiscussed in Section 1.9. They are birational to P2. A smooth quarticX4 ⊂ P3 is an example of a K3 surface (see Section 1.10). We havePic(X4) = Zr, with r between 1 and 20. They are not rational and, ingeneral, do not admit nontrivial fibrations.

Cubic and quartic surfaces have a rich geometric structure, with large“hidden” symmetries. This translates into many intricate arithmeticissues.

sect:del-pezzo1.9. Del Pezzo surfaces. A smooth projective surface X with ampleanticanonical class is called a Del Pezzo surface. Standard examplesare P2 and P1 × P1. Over algebraically closed ground fields, all otherDel Pezzo surfaces Xd are obtained as blowups of P2 in 9−d ≤ 8 pointsin general position (e.g., no three on a line, no 6 on a conic). This d isthe anticanonical degree of Xd. Del Pezzo surfaces of low degree admitthe following realizations:

• d = 4: intersection of two quadrics in P4;• d = 3: hypersurface of degree 3 in P3;• d = 2: hypersurface of degree 4 in the weighted projective space

P2(1, 1, 1, 2) given by

w2 = f4(x, y, z), with deg(f4) = 4.

• d = 1: hypersurface of degree 6 in P(1, 1, 2, 3) given by

w2 = t3 + f4(x, y)t+ f6(x, y), with deg(fi) = i.

Visually and mathematically most appealing are, perhaps, cubic sur-faces with d = 3. Note that for d = 1, the anticanonical linear serieshas one base point, in particular, X1(F ) 6= ∅, over any field F .


Let us compute the geometric invariants of a Del Pezzo surface ofdegree d, expanding the Example 1.1.3. Since Pic(P2) = ZL, the hy-perplane class, we have

Pic(Xd) = ZL⊕ ZE1 ⊕ · · · ⊕ ZE9−d,

where Ei are the full preimages of the blown-up points. The canonicalclass is computed as in Example 1.1.1

KXd = −3L+ (E1 + · · ·E9−d).

The intersection pairing defines a quadratic form on Pic(Xd), withL2 = 1, L ·Ei = 0, Ei ·Ej = 0, for i 6= j, and E2

j = −1. Let Wd be thesubgroup of GL10−d(Z) of elements preserving KXd and the intersectionpairing. For d small enough there are several other classes with square−1:

L− (Ei + Ej), 2L− (E1 + · · ·+ E5), etc.

Those classes whose intersection with KXd is also −1 are called (classesof) exceptional curves, these curves are lines in the anticanonical em-bedding. Their number r(d) can be found in the table below. Wehave

−KXd = cd

r(d)∑j=1

Ej,

the sum over all exceptional curves, where cd ∈ Q can be easily com-puted, e.g., c3 = 1/9. The effective cone is spanned by the r(d) classesof exceptional curves, and the nef cone is the cone dual to Λeff(Xd)with respect to the intersection pairing on Pic(Xd). Put

eqn:alpha-cubiceqn:alpha-cubic (1.8) α(Xd) := vol (Λnef(Xd) ∩ C | (−KX , C) = 1) .

This “volume” of the nef cone has been computed in [?]:

9− d 1 2 3 4 5 6 7 8r(d) 1 3 6 10 16 27 56 240α(Xd) 1/6 1/24 1/72 1/144 1/180 1/120 1/30 1

Given a Del Pezzo surface over a number field, the equations of thelines can be computed effectively. For example, this is easy to see forthe diagonal cubic surface

x30 + x3

1 + x32 + x3

3 = 0.

16 YURI TSCHINKEL

Writing

x3i + x3

j =3∏r=1

(xi + ζr3xj) = x3` + x3

k =3∏r=1

(x` + ζr3xk)0,

with i, j, k, l ∈ [0, . . . , 3], and permuting indices we get all 27 lines. Ingeneral, equations for the lines can be obtained by solving the corre-sponding equations on the Grassmannian of lines.

For 9 − d = 3, 4, 5, . . . , 8, the group Wd is the Weyl group of a rootsystem:

A1 × A2,A4,D5,E6,E7,E8,

and the root lattice itself is the orthogonal to KXd in Pic(Xd), the so-called primitive Picard group. Recent work on Cox rings of Del Pezzosurfaces established a geometric connection to Lie groups with rootsystems of the above type: a universal torsor over a Del Pezzo surfaceembeds into a certain flag variety for the simply-connected Lie groupwith the corresponding root system [Der07], [SS07].

Degenerations of Del Pezzo surfaces are also interesting and impor-tant. Typically, they arise as special fibers of fibrations, and theiranalysis is unavoidable in the theory of models over rings such as Z, orC[t]. There is a classification of singular Del Pezzo surfaces with thehelp of Dynkin diagrams [BW79]. Models of Del Pezzo surfaces overcurves are discussed in [Cor96]. Volumes of nef cones of singular DelPezzo surfaces are computed in [DJT08].

exam:sing-4 Example 1.9.1 (Degree four). Here are some examples of singular de-gree four Del Pezzo surfaces X = Q0 = 0 ∩ Q = 0 ⊂ P4, whereQ0 = x0x1 + x2

2 and Q is one of the following quadratic forms

4A1 x3x4 + x22

3A1 x2(x1 + x2) + x3x4

2A1 + A2 x1x2 + x3x4

A1 + A3 x23 + x4x2 + x2

0

2A1 + A3 x20 + x3x4

A3 x23 + x4x2 + (x0 + x1)2

D4 x23 + x4x1 + (x0 + x2)2

D4 x23 + x4x1 + x2

0

D5 x1x2 + x0x4 + x23

D5 x21 + x1x2 + x0x4 + x2

3.

For more details see [DP80].


exam:sing-3 Example 1.9.2 (Cubics). Here are some singular cubic surfacesX ⊂ P3,given by the vanishing of the corresponding cubic form:

4A1 x0x1x2 + x1x2x3 + x2x3x0 + x3x0x1

2A1 + A2 x0x1x2 = x23(x1 + x2 + x3)

2A1 + A3 x0x1x2 = x23(x1 + x2)

A1 + 2A2 x0x1x2 = x1x23 + x3

3

A1 + A3 x0x1x2 = (x1 + x2)(x23 − x2

1)A1 + A4 x0x1x2 = x2

3x2 + x3x21

A1 + A5 x0x1x2 = x31 − x2

3x2

3A2 x0x1x2 = x33

A4 x0x1x2 = x32 − x3x

21 + x2

3x2

A5 x33 = x3

1 + x0x23 − x2

2x3

D4 x1x2x3 = x0(x1 + x2 + x3)2

D5 x0x21 + x1x

23 + x2

2x3

E6 x33 = x1(x1x0 + x2

2).

In practice, most geometric questions are easier for smooth surfaces,while most arithmetic questions turn out to be easier in the singularcase. For a survey on arithmetic problems on rational surfaces, seeSections 2.4 and 3.4, as well as [MT86].

sect:K31.10. K3 surfaces. Let X be a smooth projective surface with trivialcanonical class. There are two possibilities: X could be an abeliansurface or a K3 surface. In the latter case, X is simply-connectedand h1(X,OX) = 0. The Picard group Pic(X) of a K3 surface X is atorsion-free Z-module of rank ≤ 20 and the intersection form on Pic(X)is even, i.e., the square of every class is an even integer. K3 surfaces ofwith polarizations of small degree can be realized as complete intersec-tions in projective space. The most common examples are K3 surfacesof degree 2, given explicitly as double covers X → P2 ramified in acurve of degree 6; or quartic surfaces X ⊂ P3.

exam:rank-k3 Example 1.10.1. The Fermat quartic

x4 + y4 + z4 + w4 = 0

has Picard rank 20 over Q(√−1). The surface X given by

xy3 + yz3 + zx3 + w4 = 0

has Pic(X) = Z20 over Q (see [Ino78] for more examples). The surface

w(x3 +y3 +z3 +x2z+xw2) = 3x2y2 +4x2yz+x2z2 +xy2z+xyz2 +y2z2

has geometric Picard rank 1, i.e., Pic(XQ) = Z [vL07].

18 YURI TSCHINKEL

Other interesting examples arise from abelian surfaces as follows:Let

ι : A → Aa 7→ −a

be the standard involution. Its fixed points are the 2-torsion pointsof A. The quotient A/ι has 16 singularieties (the images of the fixedpoints). The minimal resolution of these singularities is a K3 surface,called a Kummer surface. There are several other finite group actionson abelian surfaces such that a similar construction results in a K3surface, a generalizied Kummer surface (see [Kat87]).

The nef cone of a polarized K3 surface (X, g) admits the followingcharacterization: h is ample if and only if (h,C) > 0 for each classC with (g, C) > 0 and (C,C) ≥ −2. The Torelli theorem implies anintrinsic description of automorphisms: every automorphism of the lat-tice Pic(X) preserving the nef cone arises from an automorphisms ofX. There is an extensive literature devoted to the classification of aris-ing automorphism groups [Nik81], [Dol08]. These automorphisms giveexamples of interesting algebraic dynamical systems [McM02], [Can01].they can be used to propagate rational points and curves [BT00], andto define canonical heights [Sil91], [Kaw08].

sect:3folds1.11. Threefolds. The classification of smooth Fano threefolds wasa major achievement completed in the works of Iskovskikh [Isk79],[IP99a], and Mori–Mukai [MM86]. There are more than 100 families.Among them, for example, cubics X3 ⊂ P4, quartics X4 ⊂ P4 or doublecovers of W2 → P3, ramified in a surface of degree 6. Many of these va-rieties, including the above examples, are not rational. Unirationalityof cubics can be seen directly: projecting from a line on X3 we get a cu-bic surface fibration, which splits after base change. The nonrationalityof cubics was proved in [CG72] using intermediate Jacobians. Nonra-tionality of quartics was proved by establishing birational rigidity, i.e.,showing triviality of the group of birational automorphisms, via ananalysis of maximal singularities of such maps [IM71]. This techniquehas been substantially developed within the Minimal Model Program(see [Isk01], [Puk98], [Puk07], [Che05]) Some quartic threefolds are alsounirational, e.g., the diagonal, Fermat type, quartic

4∑i=0

x4i = 0.


It is expected that the general quartic is not unirational. However, itadmits an elliptic fibration: fix a line l ∈ X4 ⊂ P4 and consider a planein P4 containing this line, the residual plane curve has degree threeand genus 1. A general double cover W2 does not admit an elliptic orabelian fibration, even birationally [?].

sect:holo1.12. Holomorphic symplectic varieties. Let X be a smooth pro-jective simply-connected variety. It is called holomorphic symplectic ifit carries a unique, modulo constants, nondegenerate holomorphic two-form. Typical examples are K3 surfaces X and their Hilbert schemesX [n] of zero-dimensional length-n subschemes. Another example is thevariety of lines of a smooth cubic fourfold, it is deformation equivalentto X [2] of a K3 surface [?].

These varieties are interesting for the following reasons:

• the symplectic forms allows to define a quadratic form on Pic(X),the Beauville–Bogomolov form;• there is a local Torelli theorem;• there is a conjectural characterization of the ample cone, at least

in dimension 4 [?].

Further, there are examples with (Lagrangian) abelian fibrations overPn or with infinite endomorphisms, resp. birational automorphisms,which are interesting for arithmetic and algebraic dynamics.

sect:nonclosed1.13. Geometry over nonclosed fields. Issues: descent, effectivecomputation of invariants over the groundfield.

Example 1.13.1. The Picard group of cubic surfaces may be smallerover nonclosed fields: for X given by

x30 + x3

1 + x32 + x3

3 = 0

Pic(XQ) = Z4. It has a basis e1, e2, e3, e4. such that Λeff(X) is spannedby

e2, e3, 3e1 − 2e3 − e4, 2e1 − e2 − e3 − e4, e1 − e4,

4e1 − 2e2 − 2e3 − e4, e1 − e2, 2e1 − 2e2 − e4, 2e1 − e3

(see [PT01]).

Computing Picard groups[VAZ], [Zar08],Effective

20 YURI TSCHINKEL

Example 1.13.2. Let X be a K3 surface over a number field Q. Fix amodel X over Z. For primes p of good reduction we have an injection

Pic(XQ) → Pic(XFp).

The rank ρp of Pic(XFp) is always even. In some examples, ρp can becomputed by counting points over Fpr , for several r, and by using theWeil conjectures.

This local information can sometimes be used to determine the rankρQ of Pic(XQ). Let p, q be distinct primes of good reduction such thatρp, ρq ≤ 2 and the discriminants of the lattices Pic(XFp), Pic(XFq) donot differ by a square of a rational number. Then ρQ = 1.

2. Existence of pointssect:existence

2.1. Projective spaces and their forms. Let F be a field and F analgebraic closure of F . A projective space over F has many rationalpoints: they are dense in Zariski topology and in the adelic topology.Varieties F -birational to a projective space inherit these properties.

Over nonclosed fields F , projective spaces have forms, so calledBrauer–Severi varieties. These are isomorphic to Pn over F but notnecessarily over F . They can be classified via the nonabelian cohomol-ogy group H1(F,Aut(Pn)), where Aut(Pn) = PGLn+1 is the group ofalgebraic automorphisms of Pn. The basic example is a conic C ⊂ P2,e.g.,

eqn:legendreeqn:legendre (2.1) ax2 + by2 = cz2, with a, b, c ∈ N.It is easy to verify solvability of this equation modulo p, for p - abc (cf.Theorem ??). Legendre proved that (2.1) has nontrivial solutions inZ if and only if it has nontrivial solutions modulo p, for all primes p.This is an instance of a local-to-global principle that will be discussedin Section ??.

Checking solvability modulo p is a finite problem which gives a finiteprocedure to verify solvability in Z. Actually, Legendre’s proof provideseffective bounds for the size of the smallest solution, e.g.,

max(|x|, |y|, |z|) ≤ abc,

which gives another approach to checking solvability - try all x, y, z ∈ Nsubject to the inequality. If C(Q) 6= ∅, then the conic is Q-isomorphicto P1: draw lines through a Q-point in C.

In general, forms of Pn over number fields satisfy the local-to-globalprinciple. Moreover, Brauer–Severi varieties with at least one F -rationalpoint are split over F , i.e., isomorphic to Pn over F . It would be useful


to have a routine (in Magma) that would check efficiently whether ornot a Brauer–Severi variety of small dimension over Q, presented byexplicit equations, is split, and to find the smallest solution.

sect:hyper2.2. Hypersurfaces. Algebraically, the simplest examples of varietiesare hypersurfaces, defined by a single homogeneous equation f(x) = 0.Many classical diophantine problems reduce to the study of rationalpoints on hypersurfaces.

Theorem 2.2.1 (Chevalley-Warning, Abh. M. Sem. Hamb. (1936)).Let X = Xf ⊂ Pn be a hypersurface over a finite field F given by theequation f(x) = 0. If deg(f) ≤ n then X(F ) 6= ∅.

Proof. We reproduce a textbook argument [BS66], for F = Fp.Step 1. Consider the δ-function

p−1∑x=1

xd =

−1 mod p if p− 1 | d0 mod p if p− 1 - d

Step 2. Apply it to a (not necessarily homogeneous) polynomial φ ∈Fp[x0, . . . , xn], with deg(φ) ≤ n(p− 1). Then∑

x0,...,xn

φ(x0, . . . , xn) = 0 mod p.

Indeed, for monomials, we have∑x0,...,xm

xd11 · · · xdnn =

∏(∑

xdjj ), with d0 + . . .+ dn ≤ n(p− 1).

For some j, we have 0 ≤ dj < p− 1.

Step 3. For φ(x) = 1−f(x)p−1 we have deg(φ) ≤ deg(f) · (p−1). Then

N(f) := #x | f(x) = 0 =∑

x0,...,xn

φ(x) = 0 mod p,

since deg(f) ≤ n.

Step 4. The equation f(x) = 0 has a trivial solution. It follows that

N(f) > 1 and Xf (Fp) 6= ∅.

Jacobi sums ...

22 YURI TSCHINKEL

Theorem 2.2.2. [Esn03] If X is a Fano variety over a finite field Fqthen

X(Fq) 6= ∅.

Now we pass to the case in which F = Q. Given a form f ∈Z[x0, . . . , xn], homogeneous of degree d, we ask how many solutionsx = (x0, . . . , xn) ∈ Zn+1 to the equation f(x) = 0 should we expect?Primitive solutions with gcd(x0, . . . , xn) = 1, up to diagonal multipli-cation with ±1, are in bijection to rational points on the hypersurfaceXf ⊂ Pn. We have |f(x)| = O(Bd), for ‖x‖ := maxj(|xj|) ≤ B. Wemay argue that f takes values 0, 1, 2, . . . with equal probability, so thatthe the probability of f(x) = 0 is B−d. There are Bn+1 “events” with‖x‖ ≤ B. In conclusion, we expect Bn+1−d solutions with ‖x‖ ≤ B.There are three cases:

• n+1 < d: as B →∞ we should have fewer and fewer solutions,and, eventually, none!• n + 1 − d: this is the stable regime, we get no information in

the limit B →∞;• n+ 1− d: the expected number of solutions grows.

We will see many instances when this heuristic reasoning fails. How-ever, it is reasonable, as a first approximation, at least when n+1−d0.

Diagonal hypersurfaces have attracted the attention of computa-tional number theorists (see http://euler.free.fr). A sample isgiven below:

exam:many Example 2.2.3.

• There are no rational points (with non-zero coordinates) on theFano 5-fold x6

0 =∑6

j=1 x6j with height ≤ 2.6 · 106.

• There are 12 (up to signs and permutations) rational points onx6

0 =∑7

j=1 x6j of height ≤ 105 (with non-zero coordinates).

• The number of rational points (up to signs, permutations andwith non-zero coordinates) on the Fano 5-fold x6

0+x61 =

∑6j=2 x

6j

of height ≤ 104 (resp. 2 · 104, 3 · 104) is 12 (resp. 33, 57).

It is visible, that it is difficult to generate solutions when the n− dis small. On the other hand, we have a the following theorem:


thm:birch Theorem 2.2.4. [Bir62] If n ≥ (deg(f)− 1) · 2deg(f), and f is smooth,then Xf satisfies the Hasse principle and the number N(f,B) of solu-tions x = (xi) with max(|xi|) ≤ B is

N(f,B) ∼∏p

τp · τ∞Bn+1−d, as B →∞,

where τp, τ∞ are the p-adic, resp. real, densities.

Further developments:

• asymptotic formulas and weak approximation• better bounds for n for small deg(f)• works over Fq[t]

Now we assume that X = Xf is a hypersurface over a function fieldin one variable F = C(t). We have

Theorem 2.2.5. If deg(f) ≤ n then Xf (C(t)) 6= ∅.

Proof. It suffices to count paramenters: Insert xj = xj(t) ∈ C[t], ofdegree e, into

f =∑J

fJxJ = 0,

with |J | = deg(f). This gives a system of e · deg(f) + const equationsin e(n + 1) variables. This system is solvable for e 0, provideddeg(f) ≤ n.

sect:decide2.3. Decidability. Hilbert’s 10th problem.

Matiyasevich (1970), Matiyasevich-Robinson (1975)

Theorem 2.3.1. Let f ∈ Z[t, z1, . . . , xn] be polynomial. The set oft ∈ Z such that f(t, . . . , xn) = 0 is solvable in Z is not decidable, i.e.,there is no algorithm to decide whether or not a diophantine equationis solvable in integers.

Theorem 2.3.2. [?] The set of t ∈ Z such that ft = 0 has infinitelymany primitive solutions is algorithmically random 1.

1The author’s abstract: “One normally thinks that everything that is true is true for a

reason. I’ve found mathematical truths that are true for no reason at all. These mathematical

truths are beyond the power of mathematical reasoning because they are accidental and random.

Using software written in Mathematica that runs on an IBM RS/6000 workstation, I constructed

a perverse 200-page algebraic equation with a parameter t and 17,000 unknowns. For each whole-

number value of the parameter t, we ask whether this equation has a finite or an infinite number

of whole number solutions. The answers escape the power of mathematical reason because they

are completely random and accidental.”

24 YURI TSCHINKEL

Decidability over other rings

• Yes: Z[S−1], for some sets S of primes, of density 1 (Poo-nen/Shlapentokh 2003)• Yes: Fp, Qp, R, C• Unknown: OK , number fields K (including Q)• Unknown: C(t)• No: Fq[t], C(t1, . . . , tn) with n ≥ 2

sect:obstruct2.4. Obstructions. As we have seen in Section 2.3, there is no hopeof finding an algorithm which would determine the solvability of adiophantine equation in integers, i.e., there is no algorithm to testfor the existence of integral points on quasi-projective varieties. Thecorresponding question for homogeneous equations, i.e., for rationalpoints, is still open. It is reasonable to expect that at least for someclasses of algebraic varieties, for example, for Del Pezzo surfaces, theexistence question can be answered. In this section we survey somerecent results in this direction.

Let XB be a scheme over a base scheme B. We are looking forobstructions to the existence of points X(B), i.e., sections of the struc-ture morphism X → B. Each morphism B′ → B gives rise to abase-change diagram, and each section x : B → X provides a sectionx′ : B′ → XB′ .

X

XB′oo

X XB′oo

B B′oo B

x

OO

B′oo

x′

OO

This gives rise to a local obstruction, since it is sometimes easier tocheck that XB′(B

′) = ∅. In practice, B could be a curve and B′ a cover,or an analytic neighborhood of a point on B. In the number-theoreticcontext, B = Spec (F ) and B′ = Spec (Fv), where v is a valuation ofthe number field F and Fv the v-adic completion of F . One says thatthe local-global principle, or the Hasse principle, holds, if the existenceof F -rational points is implied by the existence of v-adic points in allcompletions.

exam:hasse Example 2.4.1. The Hasse principle holds for:

(1) smooth quadrics X2 ⊂ Pn;(2) Brauer–Severi varieties;(3) Del Pezzo surfaces of degree ≥ 5;


(4) Chatelet surfaces y2 − az2 = f(x0, . . . , xn), where f is an irre-ducible polynomial of degree ≤ 4 [CTSSD87b];

(5) hypersurfaces Xd ⊂ Pn, for n d (see Theorem 4.8.1).

The Hasse principle may fail for cubic curves, e.g.,

3x3 + 4y3 + 5z3 = 0.

In topology, there is a rich obstruction theory to the existence ofsections. An adaptation to algebraic geometry is formulated as follows:Let C be a contravariant functor from the category of schemes over abase scheme B to the category of abelian groups. Applying the functorC to the diagrams above, we have

C(X) // C(XB′) C(X) //

x

C(XB′)

x′

C(B) //

OO

C(B′)

OO

C(B) // C(B′)

If for all sections x′, the image of x′ in C(B′) is nontrivial in thecokernel of the map C(B) → C(B′), then we have a problem, i.e., anobstruction to the existence of B-points on X. So far, this is still aversion of a local obstruction. However, a global obstruction may arise,when we vary B′.

We are interested in the case when B = Spec (F ), for a number fieldF , with B′ ranging over all completions Fv. A global obstruction ispossible whenever the map

C(Spec (F ))→∏v

C(Spec (Fv))

has a nontrivial cokernel. What are sensible choices for C? Basic con-travariant functors on schemes are C(−) := Hi

et(−,Gm). For i = 1,we get the Picard functor, introduced in Section 1.1. However, byHilbert’s theorem 90, H1

et(Spec (F ),Gm) = 0, for all fields F , and thiswon’t generate an obstruction. For i = 2, we get the (cohomological)Brauer group Br(X) = H2

et(X,Gm), classifying sheaves of central sim-ple algebras over X, modulo equivalence (see [?, Chapter 4]). By classclass field theory, we have an exact sequence

ean:brauerean:brauer (2.2) 0→ Br(F )→⊕v

Br(Fv)Pv invv−→ Q/Z→ 0,

where invv : Br(Fv)→ Q/Z is the local invariant. We apply it to thediagram and obtain:

26 YURI TSCHINKEL

Br(XF ) //

x

⊕v Br(XFv)

(xv)v

0 // Br(F ) //⊕

v Br(Fv)Pv invv // Q/Z // 0,

Define

eqn:hasse-prineqn:hasse-prin (2.3) X(AF )Br := ∩A∈Br(X)(xv)v ∈ X(AF ) |∑v

inv(A(xv)) = 0.

Let X(F ) be the closure of X(F ) in X(AF ), in the adelic topology. One

says that X satisfies weak approximation over F if X(F ) = X(AF ). Wehave

X(F ) ⊂ X(F ) ⊂ X(AF )Br ⊂ X(AF ).

From this we derive the Brauer–Manin obstruction to the Hasse prin-ciple and weak approximation:

• if X(AF ) 6= ∅ but X(AF )Br = ∅ then X(F ) = ∅, i.e., the Hasseprinciple fails;• if X(AF ) 6= X(AF )Br then weak approximation fails.

Del Pezzo surfaces of degree ≥ 5 satisfy the Hasse principle and weakapproximation. Arithmetically most interesting are Del Pezzo surfacesof degree 4, 3, and 2: these may fail the Hasse principle:

• deg = 4: z2 + w2 = (x2 − 2y2)(3y2 − x2) [Isk71];• deg = 3: 5x3 + 12y3 + 9z3 + 10w3 = 0 [CG66];• deg = 2: w2 = 2x4 − 3y4 − 6z4 [KT04].

One says that the Brauer–Manin obstruction to the existence of ra-tional point is the only one, if X(AF )Br 6= ∅ implies that X(F ) 6= ∅.This holds for:

(1) abelian varieties over a number field F with finite Tate–Shafarevichgroup [Man71];

(2) principal homogeneous spaces for a connected linear algebraicgroup;

(3) Del Pezzo surfaces of degree ≥ 3 admitting a conic bundle struc-ture defined over the groundfield F .

(4) conjecturally(!), for all geometrically rational surfaces.

descent, torsor formalismSpecialization: x ∈ X(F ) 7→ [Tx] ∈ H1

et(F, T ), a finite set. This givesa partition:

X(F ) = ∪α∈H1et(F,T )πα(Tα(F )),


where Tα are the descent varieties.Transcendental obstructions: [Har96]

sect:descent2.5. Descent. Abelian descent

Nonabelian descent [Har08], [Sko99], [HS05], [HS02]Bjorn’s example [Poo08]For detailed surveys of the theory of the Brauer-Manin obstruction

are [Pey05], [Har04].sect:effect

2.6. Effectivity. In light of the discussion in Section 2.3 it is impor-tant to know whether or not the Brauer–Manin obstruction can be com-puted, effectively in terms of the coefficients of the defining equations.There is an extensive literature on computations of the Brauer–Maninobstruction for curves [].

Effective computability of the Brauer-Manin obstruction for all DelPezzo surfaces over number fields has been proved in [KT08]. The mainsteps are as follows:

(1) Computation of the equations of the exceptional curves and ofthe action of the Galois group of the splitting field G on thesecurves as in Section 1.13. One obtains the exact sequence ofG-modules

0→ Relations→ ⊕ZEj → Pic(X)→ 0.

(2) We have

Br(X)/Br(F ) = H1(G,Pic(X)).

Using the equations for exceptional curves and functions realiz-ing relations between the curves classes in the Picard group onecan compute explicitly Azumaya algebras Ai representing theclasses of Br(X)/Br(F ).

(3) The local points S(Fv) can be effectively decomposed into a fi-nite union of subsets such that each Ai is constant on each ofthese subsets. This step uses an effective version of the arith-metic Hilbert Nullstellensatz.

(4) It remains to compute the local invariants.

2.7.

Example 2.7.1. [HBM04] Let a, b ∈ Z be coprime such that a = ±bmod 9. Then the cubic surface

x30 + 2x3

1 + ax32 + bx3

3 = 0

has a Q-rational point.

28 YURI TSCHINKEL

The proof combines a deep result in the theory of elliptic curves[?], namely, the existence of nontrivial Heegner points on the curvex3

0 + 2x31 = py3, for any prime p = 2 mod 9, with a result in analytic

number theory on representations of primes p by the cubic form ax32 +

bx33.

3. Density of pointssect:densitysect:lang

3.1. Lang’s conjecture. One of the main principles underlying arith-metic geometry is the expectation that the trichotomy in the classifi-cation of algebraic varieties via the Kodaira dimension in Section 1.2has an arithmetic manifestation. The broadly accepted form of this is

conj:lang Conjecture 3.1.1 (Lang’s conjecture). Let X be a variety of generaltype, i.e., a smooth projective variety with ample canonical class, de-fined over a number field F . Then X(F ) is not Zariski dense.

What about a converse? The obvious necessary condition for Zariskidensity of rational points, granted Conjecture ??, is that X does notdominate a variety of general type. This condition is not enough, aswas shown in [?]: there exist surfaces which do not dominate curves ofgeneral type but which have etale covers dominating curves of generaltype. By the Chevalley–Weil theorem (see Section ??), these coverswould have a dense set of rational points, over some finite extension ofthe ground field, contradicting Conjecture ??.

The formulation of a converse statementFor a detailed discussion of the geometric considerations related to

potential density and Campana’s program see [Abr08].sect:dense-fixed

3.2. Zariski density over fixed fields. Here we discuss Zariski den-sity of rational points in the “unstable” situation, when the density ofpoints is governed by subtle number-theoretical properties, rather thangeometric considerations. We have the following fundamental result:

thm:pot-curve Theorem 3.2.1. Let C be a smooth curve of genus g = g(C) over anumber field F . Then

• if g = 0 and C(F ) 6= ∅ then C(F ) is Zariski dense;• if g = 1 and C(F ) 6= ∅ then C(F ) is an abelian group (the

Mordell-Weil group) and there is a constant cF (independent ofC) bounding the order of the torsion subgroup C(F )tors of C(F )[Maz77], [Mer96]; in particular, if there is an F -rational pointof infinite order then C(F ) is Zariski dense;• if g ≥ 2 then C(F ) is finite [Fal83], [Fal91].


In fact, assuming the Lang-Vojta conjecture, one obtains uniformbounds for #C(F ), if g(C) ≥ 2 [CHM97b], [CHM97a]. EXPAND...

In higher dimensions we have:

thm:pot-de Theorem 3.2.2. Let X be an algebraic variety over a number field F .Assume that X(F ) 6= ∅ and that X is one of the following

• X is a Del Pezzo surface of degree 2 and has a point on thecomplement to exceptional curves;• X is a Del Pezzo surface of degree ≥ 3;• X is a Brauer-Severi variety.

Then X(F ) is Zariski dense.

The proof of the first claims can be found in [Man86, p. ??].

rema:de1 Remark 3.2.3. Let X/F be a Del Pezzo surface of degree 1 (it alwayscontains an F -rational point, the base point of the anticanonical linearseries) or a conic bundle X → P1, with X(F ) 6= ∅. It is unknownwhether or not X(F ) is Zariski dense.

Theorem 3.2.4. [Elk88] Let X ⊂ P3 be the quartic K3 surface givenby

eqn:xxeqn:xx (3.1) x40 + x4

1 + x42 = x4

3.

Then X(Q) is Zariski dense.

The trivial solutions (1 : 0 : 0 : 1) etc are easily seen. The smallestnontrivial solution is ...

Example 3.2.5. [EJ06] Let X ⊂ P3 be the quartic given by

x4 + 2y4 = z4 + 4t4.

The obvious Q-rational points are given by y = t = 0 and x = ±z.The next smallest solution is

14848014 + 2 · 12031204 = 11694074 + 4 · 11575204.sect:potential

3.3. Potential density: techniques. Here is a (short) list of possi-ble strategies to propagate points:

• use the group of automorphisms Aut(X), if it is infinite;• try to find a dominant map X → X where X satisfies potential

density (for example, try to prove unirationality);• try to find a fibration structure X → B where the fibers satisfy

potential density in some uniform way (that is, the field exten-sions needed to insure potential density of the fibers Vb can beuniformly controlled).

30 YURI TSCHINKEL

In particular, it is important for us keep track of minimal conditionswhich would insure Zariski density of points on varieties. A fundamen-tal result is

exem:conic-bundles Example 3.3.1. Let π : X → P1 be a conic bundle, defined over a fieldF . Then rational points on X are potentially dense. Indeed, by Tsen’stheorem, π has section s : P1 → X (which is defined over some finiteextension F ′/F ), each fiber has an F ′-rational point and it suffices toapply Theorem 3.2.1. Potential density for conic bundles over higher-dimensional bases is an open problem.

If X is an abelian variety then there exists a finite extension F ′/Fand a point P ∈ X(F ′) such that the cyclic subgroup of X(F ′) gener-ated by P is Zariski dense.

Example 3.3.2. If π : X → P1 is a Jacobian nonisotrivial ellipticfibrations (π admits a section and the j-invariant is nonconstant), thenpotential density follows from a strong form of the Birch/Swinnerton-Dyer conjecture [GM97], [Man95]. The key problem is to control thevariation of the root number (the sign of the functional equation of theL-functions of the elliptic curve) (see [GM97]).

On the other hand, rational points on the elliptic fibration .... arenot potentially dense [CTSSD97].

exam:geome-ell Example 3.3.3. One geometric approach to Zariski density of rationalpoints on (certain) elliptic fibration can be summarized as follows:

Case 1. Let π : X → P1 be a Jacobian elliptic fibration and e :P1 → X its zero-section. Suppose that we have another section swhich is nontorsion in the Mordell-Weil group of X(F (B)). Then aspecialization argument implies that the restriction of the section toinfinitely many fibers of π gives a nontorsion point in the Mordell-Weilgroup of the corresponding fiber (see [Ser90], 11.1). In particular, X(F )is Zariski dense.

Case 2. Suppose that π : X → P1 is an elliptic fibration with amultisection M (an irreducible curve surjecting onto the base P1). Aftera basechange X×P1 M →M the elliptic fibration acquires the identitysection Id (the image of the diagonal under M ×P1 M → V ×P1 M) anda (rational) section

τM := dId− Tr(M ×P1 M),


where d is the degree of π : M → P1 and Tr(M ×P1 M) is obtained(over the generic point) by summing all the points of M ×P1 M . Wewill say that M is nontorsion if τM is nontorsion.

If M is nontorsion and if M(F ) is Zariski dense then the same holdsfor X(F ) (see [BT99]).

sect:pot-dense-res3.4. Potential density for surfaces.

exem:brs-fibr Example 3.4.1. Let π : X → C be a fibration with generic fiber aBrauer-Severi variety or a Del Pezzo surface of degree ≥ 2. Theorem ??implies that π has a section. Potential density for X follows fromTheorem 3.2.2.

rema:ab-fibr Remark 3.4.2. An argument similar to Example ?? works for abelianfibrations. The difficulty here is to formulate some simple geometricconditions insuring that a (multi)section leads to points which are notonly of infinite order in the Mordell-Weil groups of the correspondingfibers, but in fact generate Zariski dense subgroups.

By Theorem 3.2.1, potential density holds for curves of genus g ≤ 1.It holds for surfaces which become rational after a finite extension of theground field, e.g., for all Del Pezzo surfaces. The classification theoryin dimension 2 gives us the following list of surfaces of intermediatetype:

• abelian surfaces;• bielliptic surfaces;• Enriques surfaces;• K3 surfaces.

Potential density for the first two classes follows from Theorem 3.2.2.The classification of Enriques surfaces X implies that either Aut(X) isinfinite or X is dominated by a K3 surface X with Aut(X) infinite [?].Thus we are reduced to the study of K3 surfaces.

Theorem 3.4.3. [BT00] Let X be a K3 surface over any field of char-acteristic zero. If X is elliptic or admits an infinite group of automor-phisms then rational points on X are potentially dense.

32 YURI TSCHINKEL

Sketch of the proof. The main difficulty is to find sufficiently nondegen-erate rational or elliptic multisections of the elliptic fibration X → P1.These are produced using deformation theory. One starts with specialK3 surfaces which have rational curves Ct ⊂ Xt in the desired homol-ogy class (for example, Kummer surfaces) and then deforms the pair.Actually, this deformation technique has to be applied to twists of theoriginal elliptic surface.

rema:genk3 Remark 3.4.4. General K3 surfaces X have (geometric) Picard num-ber one. In particular, Aut(X) is finite and there are no elliptic fibra-tions. Potential density of rational points is an open problem.

Example 3.4.5. A smooth hypersurface X ⊂ P1 × P1 × P1 of bi-degree(2, 2, 2) is a K3 surface with Aut(X) infinite.

exem:k33 Example 3.4.6. Every smooth quartic surface S4 ⊂ P3 which containsa line is an elliptic K3 surface. Indeed, let M be this line and assumethat both S4 and M are defined over a number field F . Consider the 1-parameter family of planes P2

t ⊂ P3 containing M . The residual curvein the intersection P2

t ∩ S4 is a plane cubic intersecting M in 3 points.This gives a fibration π : S4 → P1 with a rational tri-section M .

To apply the strategy of Section 6.1 we need to insure that M isnontorsion. A sufficient condition, satisfied for generic quartics S4, isthat the restriction of π to M ramifies in a smooth fiber of π : X → P1.Under this condition X(F ) is Zariski dense.

LATERWe now collect some results and open problems concerning the arith-

metic of K3 surfaces.Existence:Swinnerton-Dyer/SlaterZariski density

Theorem 3.4.7. [Elk88] Let X ⊂ P3 be the quartic K3 surface givenby

eqn:xxeqn:xx (3.2) x40 + x4

1 + x42 = x4

3.

Then X(Q) is Zariski dense.


The trivial solutions (1 : 0 : 0 : 1) etc are easily seen. The smallestnontrivial solution is ... Geometrically, over Q, the surface given by(3.2) is a Kummer surface, with many elliptic fibrations.

Theorem 3.4.8. [HT00] Let X ⊂ P3 be a quartic K3 surface con-taining a line defined over a field F . If X is general, then X(F ) isZariski dense. In all cases, there exists a finite extension F ′/F suchthat X(F ′) is Zariski dense.

Theorem 3.4.9. [?] Let X be any elliptic K3 surface over any fieldF . Then rational points are potentially dense.

sect:pot33.5. Potential density in dimension ≥ 3. Potential density holdsfor unirational varieties. Classification of (smooth) Fano threefolds andthe detailed study of occuring families implies unirationality for all butthree cases:

• X4: quartics in P4;• V1: double covers of a cone over the Veronese surface in P5

famified in a surface of degree 6;• W2: double covers of P3 ramified in a surface of degree 6.

We now sketch the proof of potential density for quartics from [HT00],the case of V1 is treated by similar techniques in [BT99].

The threefold X4 contains a 1-parameter family of lines. Choose aline M (defined over some extension of the groundfield, if necessary)and consider the 1-parameter family of hyperplanes P3

t ⊂ P4 containingM . The generic hyperplane section St := P3

t ∩X4 is a quartic surfacewith a line. Now we would like to argue as in the Example 3.4.6.We need to make sure that M is nontorsion in St for a dense set oft ∈ P1. This will be the case for general X4 and M . The analysis of allexceptional cases requires care.

Remark 3.5.1. It would be interesting to have see (nontrivial) exam-ples of Calabi-Yau threefolds with a dense set of rational points.

Theorem 3.5.2. [?] Let X be a K3 surface over a field F , of degree2d. Then rational points on X [n] are potentially dense.

The proof relies on the existence of an abelian fibration

Y := X [n] → Pn,

34 YURI TSCHINKEL

with a nontorsion multisection which has a potentially dense set ofrational points. Numerically, such fibrations are predicted by square-zero classes in the Picard group Pic(Y ), with respect to the Beauville–Bogomolov form (see Section 1.12). Geometrically, the fibration is thedegree n Jacobian fibration associated to hyperplane sections of X.

Theorem 3.5.3. [AV07] Let Y be the Fano variety of lines on a generalcubic fourfold X3 ⊂ P5 over a field of characteristic zero. Then rationalpoints on Y are potentially dense.

Sketch of proof. The key tool is a rational endomorphism φ : Y → Yanalyzed in [?]: let l on X3 ⊂ P5 be a general line and P2

l ⊂ P5 theunique plane everywhere tangent to l. Let [l] ∈ Y be the correspondingpoint and put φ([l]) := [l′], where l′ is the residual line in X3.

Generically, one can expect that the orbit φn([l])n∈N is Zariskidense in Y . This was proved by Amerik and Campana in [AC08], overuncountable ground fields. Over countable fields, one faces the diffi-culty that the countably many exceptional loci could cover all algebraicpoints of Y . Amerik and Voisin were able to overcome this obstacle overnumber fields. Rather than proving density of φn([l])n∈N they findsurfaces Σ ⊂ Y , birational to abelian surfaces, whose orbits are densein Y . The main effort goes into showing that one can choose sufficientlygeneral Σ defined over Q, provided that Y is sufficiently general andstill defined over a number field. In particular, Y has geometric Picardnumber one. A case by case geometric analysis excludes the possibilitythat the Zariski closure of φn(Σ)n∈N is a proper subvariety of F .

Theorem 3.5.4. [HT] Let Y be the variety of lines of a cubic fourfoldX3 ⊂ P5 which contains a cubic scroll T . Assume that the hyperplanesection of X3 containing T has exactly 6 double points in linear generalposition and that X3 does not contain a plane. If X3 and T are definedover a field F then Y (F ) is Zariski dense.

rema:uni-para Remark 3.5.5. In higher dimensions, (smooth) hypersurfaces Xd ⊂Pn of degree d represent a major challenge. The circle method workswell when

n 2d

while the geometric methods for proving unirationality require at leasta super-exponential growth of n (see [?] for a contruction of a unira-tional parametrization).


3.6. Density in analytic topologies. Automorphisms φ : X → Xgive rise to interesting dynamical systems. Again, K3 surfaces serveas basic examples. The forward orbits φn(x)n∈N of real points cangenerate beautiful pictures in X(R) (see [McM02]).

3.7. Approximation.

Example 3.7.1. [Wit04] Let E → P1 be the elliptic fibration given by

y2 = x(x− g)(x− h) where g(t) = 3(t− 1)3(t+ 3) and h = g(−t).Its minimal proper regular model X is an elliptic K3 surface thatfails weak approximation. The obstruction comes from transcenden-tal classes in the Brauer group of X.

4. Counting problemssect:counting

Here we consider projective algebraic varieties X ⊂ Pn defined overa number field F . We assume that X(F ) is Zariski dense. We seek tounderstand the distribution of rational points with respect to heights.

sect:heights4.1. Heights. First we assume that F = Q. Then we can define aheight integral (respectively rational points) on the affine (respectivelyprojective space) as follows

Haffine : An(Z) = Zn → R≥0

x = (x1, . . . , xn) 7→ maxj(|xj|)H : Pn(Q) = (Zn+1

prim \ 0)/± → R>0

x = (x0, . . . , xn) 7→ maxj(|xj|).This induces heights on points of subvarieties of affine or projectivespaces. In some problems it is useful to work with equivalent norms,

e.g.,√∑

x2j instead of maxj(|xj|). Such choices are referred to as a

change of metrization. A more conceptual definition of heights andadelic metrizations is given in Section ??

Heights on products, reimbeddings, etc.

4.2. Counting functions. For a subvariety X ⊂ Pn put

N(X,B) := #x ∈ X(Q) |H(x) ≤ B.What can be said about

N(X,B), for B →∞ ?

Main questions here concern:

• (uniform) upper bounds,

36 YURI TSCHINKEL

• asymptotic formulas,• geometric interpretation of the asymptotics.

By the very definition, N(X,B) depends on the projective embeddingof X. For X = Pn over Q, with the standard embedding via the linebundle O(1), we get

N(Pn, B) ∼ 1

ζ(n+ 1)· τ∞ ·Bn+1, B →∞.

But we may also consider the Veronese re-embedding

Pn → PNx 7→ xI , |I| = d,

e.g.,

P1 → P2

(x0 : x1) 7→ (x20 : x0x1 : x2

1)

The image y0y2 = y21 has ∼ B points of height ≤ B. Similarly, the

number of rational points on height ≤ B in the O(d) embedding of Pnwill be ∼ B(n+1)/d.

More generally, if F/Q is a finite extension, put

Pn(F ) → R>0

x 7→∏

v max(|xj|v).

Theorem 4.2.1. [?]thm:schanuel

eqn:schanueleqn:schanuel (4.1) N(Pn(F ), B) ∼ hFRF (n+ 1)r1+r2−1

wF ζF (n+ 1)

(2r1(2π)r2√

disc(F )

)n+1

Bn+1,

where

• hF is the class number of F ;• RF the regulator;• r1 (resp. r2) the number of real (resp. pairs of complex) embed-

dings of F ,• disc(F ) the discriminant;• wF the number of roots of 1 in F ;• ζF the zeta function of F .

With this starting point, one may try to prove asymptotic formulas ofsimilar precision for arbitrary projective algebraic varieties X, at leastunder some natural geometric conditions. This program was initiatedin [FMT89] and it has rapidly grown in recent years.


sect:experiments4.3. Experiments. Experimental data

Elsenhans/Jahnel

4.4. Upper bounds. Pila (1995)

Theorem 4.4.1. Let X ⊂ Pn be a geometrically irreducible variety,and ε > 0. Then

N(X,B)deg(X),dim(X),ε Bdim(X)+ 1deg(X)

+ε

Based on previous work of Bombieri-Pila (1989) - geometry of num-bers.

• Pila (1996): Let X ⊂ A2 be a geometrically irreducible affinecurve. Then

#x ∈ X(Z) |H(x) ≤ B deg(X) B1

deg(X) log(B)2deg(X)+3.

• Ellenberg/Venkatesh (2005): Let X ⊂ P2 be a geometricallyirreducible curve of genus ≥ 1. Then there is a δ > 0 such that

N(X,B)deg(X),δ B2

deg(X)−δ.

Browning, Heath-Brown, Salberger (2006)

Theorem 4.4.2. Let X ⊂ Pn be a geometrically irreducible variety,and ε > 0. Then

N(X,B)deg(X),dim(X),ε

B

dim(X)− 34

+ 53√

3+ε

deg(X) = 3

Bdim(X)− 2

3+ 3

2√

deg(X)+ε

deg(X) = 4, 5Bdim(X)+ε deg(X) ≥ 6

4.5. Lower bounds. Lower bounds for Del surfaces

lem:lower Lemma 4.5.1. Let X be a smooth Fano variety over a number fieldF and Y := BlZ(X) a blowup in a smooth subvariety Z = ZF ofcodimension ≥ 2. If N(X(F ),−KX ,B) B1, for all dense Zariskiopen X ⊂ X then the same holds for Y :

N(Y (F ),−KY ,B) B1.

Proof.

In particular, split Del Pezzo surfaces Xd satisfy the lower bound ofConjecture 4.12.1

N(Xd(F ),−KXd ,B) B1.

38 YURI TSCHINKEL

Finer lower bounds, in some nonsplit cases have been proved in [?], [?]:

N(X3 (F ),B) B log(B)r−1,

if X3 is a cubic surface with at least two skew lines defined over F . Thisgives support to Conjecture 4.12.2. The following theorem providesevidence for Conjecture 4.12.1 in dimension 3.

Theorem 4.5.2. [?] Let X be a Fano threefold over a number field F0.theo:manin-3For every Zariski open subset X ⊂ X there exists a finite extensionF/F0 such that

N(X(F ),−KX ,B) B1

This relies on the classification of Fano threefolds (see [IP99b], [MM82],[MM86]). One case was missing from the classification when [?] waspublished; the Fano threefold obtained as a blowup of P1 × P1 × P1 isa curve of tri-degree (1, 1, 3) [MM03]. Lemma ?? proves the expectedlower bound in this case as well.

4.6. Finer issues.

• local or global obstructions :x2

0 + x21 + x2

2 = 0 or x30 + 4x3

1 + 10x32 + 25x3

3 = 0• singularities :

x21x

22 + x2

2x23 + x2

3x21 = x0x1x3x4 has ∼ B3/2 points of height

≤ B, on every Zariski open subsetaccumulating subvarieties :For example, x3

0 +x31 +x3

2 +x33 = 0 has ∼ B2 points on Q-lines

and and provably O(B4/3+ε) points in the complement [?]. Theexpectation is B log(B)3, over Q. Similar effects persist in higherdimensions. A quartic X4 ⊂ P4 contains a 1-parameter familyof lines, each contributing ∼ B2 to the asymptotic, while theexpectation is ∼ B. Lines on a cubic X3 ⊂ P4 are parametrizedby a surface, which is of general type. We expect ∼ B2 pointsof height ≤ B on the cubic threefold, and on each line. In [?,Theorem 11.10.11] it is shown that

Nlines(B) ∼ cB2, as B→∞,where the count is over F -rational points on lines defined overF , and the constant c is a converging sum of leading terms ofcontributions from each line of the type (4.1). In particular,each line contributes a positive density to the main term. Onthe other hand, one expects the same asymptotic ∼ B2 on thecomplement of lines, with the leading term a product of local


densities. How to reconcile this? The forced compromise isto discard such accumulting subvarieties and to hope that forsome Zariski open subset U ⊂ X, the asymptotic of points ofbounded height does reflect the geometry of X.

Xf ⊂ Pn - open questionsIf Xf is smooth, with deg(f) ≤ n and n ≥ 4 then

• Potential density : there exists a finite extension F/Q such thatX(F ) is Zariski dense;• Growth: N(Xf (F ), B) ∼ τ ·Bn+1−d;• Tamagawa number : τ =

∏v τv - product of local densities.

Hold when deg(f) = 2 and n ≥ 2, except when n = 3 and thediscriminant of the quadric is a square: B2 log(B).

Consider the variety X ⊂ P5 over Q given by

x0x1 − x2x3 + x4x5 = 0.

It is visibly a quadric hypersurface and we could apply the circlemethod as in Section ??. It is also the Grassmannian variety Gr(2, 4)and an equivariant compactification of G4

a. We could count points usingany of the structures.

Cubic surfaces X ⊂ P3 - conjectures

• Global obstruction: the Brauer-Manin obstruction to the Hasseprinciple is the only one;• Linear Growth: N(X(F ), B) ∼ c · B1 log(B)r−1, where r =

rk Pic(X);• Leading coefficient : c = αβτ , where α ∈ Q, β = |H1(Gal,Pic(X)|,

and

τ =

∫X(F )

ωK

a Tamagawa type number.

For X split over Q: α = 1/(7! · 120), β = 1 and τ =∏

p τp · τ∞ is theproduct of local densities, for almost all p:

τp = (1 +7

p+ 1)(1− 1

p)7.

Cubic surfaces X ⊂ P3 - resultsUpper bound - Heath-Brown (2002):

N(X(F ), B)ε B52/27+ε,

where X is the complement to F -lines on X

Salberger (2006): N(X(F ), B)ε B√

3+ε

40 YURI TSCHINKEL

Asymptotic - Derenthal (2005): linear growth + leading coeffient forthe E6-cubic:

x1x22 + x2x

20 + x3

3 = 0.

conj:layer Conjecture 4.6.1. Let X be a K3 surface over a number field F . LetL be a polarization, ε > 0 and Y = Y (ε, L) be the union of all F -rational curves C ⊂ X (i.e., curves that are isomorphic to P1 over F )and have L-degree ≤ 2/ε. Then

N(X,L,B) = N(Y, L,B) +O(Bε), as B →∞.

Theorem 4.6.2. [McK00] Let X → P1×P1 be a double cover ramifiedover a curve of bidegree (4, 4). Then there exists an open cone Λ ⊂Λample(X) such that for every L ∈ Λ there exists a δ > 0 such that

N(X,L,B) = N(Y, L,B) +O(B2/d−δ), as B →∞,

where d is the minimal L-degree of a rational curve on X and Y is theunion of all F -rational curves of degree d.

This theory exhibits the first layer of an arithmetic stratificationpredicted in Conjecture 4.6.1.

4.7. Upper bounds. Fibration method[BHBS06]

sect:cirle4.8. The cicle method. Let f ∈ Z[x0, . . . , xn] be a homogeneouspolynomial of degree d such that the hypersurface Xf ⊂ Pn is nonsin-gular. Let

Nf (B) := #x ∈ Zn | f(x) = 0 ‖x‖ ≤ B

be the counting function. In this section we sketch a proof of thefollowing

Theorem 4.8.1. [Bir62] Assume that n ≥ 2d(d+ 1). Thenthm:birch

eqn:circle-thetaeqn:circle-theta (4.2) Nf (B) = Θ ·Bn+1−d(1 + o(1)) B →∞,

where

Θ =∏p

τp · τ∞ > 0,

provided f(x) = 0 is solvable in Zp, for all p, and in R.

The constants τp and τ∞ admit an interpretation as local densitities;these are explained in a more conceptual framework in Section 4.14.


Substantial efforts have been put into reducing the number of vari-ables, especially for low degrees. (See HB 2006, 2007...) Another di-rection is the extension of the method to more general number fields[Ski97] and even function fields.

We now outline the main steps of the proof of the asymptotic for-mula 4.2. The first step is the introduction of a “delta”-function: forx ∈ Z we have ∫ 1

0

e2πiαxdα =

0 if x 6= 01 otherwise

Now we can write

eqn:Neqn:N (4.3) Nf (B) =

∫ 1

0

S(α)dα

where

S(α) :=∑

x∈Zn+1, ‖x‖≤B

e2πiαf(x).

The function S(α) is wildly oscillating (see Figure 4.8), with peaks atα = a/q, with small q. Indeed, the probability that f(x) is divisible byq is higher for small q, and each such term contributes 1 to S(α). Theidea of the circle method is to analyze the asymptotic of the integralin equation 4.3, for B →∞, by extracting the contributions of α closeto rational numbers a/q with small q, and finding appropriate boundsfor integrals over the remaining intervals.

[?], [HB06]More precisely, one introduces the major arcs

M :=⋃

(a,q)=1, q≤B∆

Ma,q,

where ∆ > 0 is a parameter to be specified, and

Ma,q := α | |α− a

q| ≤ B−d+δ.

The minor arcs are the complement:

m := [0, 1] \M.

The goal is to prove the bound

eqn:minoreqn:minor (4.4)

∫m

S(α)dα = O(Bn−d−ε), for some ε > 0,

42 YURI TSCHINKEL

fig:s

Figure 1. Oscillations of S(α)

and the asymptotic

eqn:majoreqn:major (4.5)

∫M

S(α)dα ∼∏p

τp · τ∞ ·Bn+1−d for B →∞.

Remark 4.8.2. Modern refinements employ “smoothed out” intervals,i.e., the delta function of an interval in the major arcts is replaced bya smooth bell curve with support in this interval. In Fourier analysis,“rough edges” translate into bad bounds on the dual side, and shouldbe avoided. An implementation of this idea, leading to savings in thenumber of variables can be found in [HB83].

There are various approaches to proving upper bounds in equa-tion 4.4, most are a variant or refinement of Weyl’s bounds (1916)[Wey16]. Weyl considered the following exponential sums

s(α) :=∑

0≤x≤B

e2πiαxd ,


The main observation is that |s(α)| is “small”, when |α− a/q| “large”.This is easy to see when d = 1; summing the geometric series we get

|s(α)| = |1− e2πiα(B+1)

1− e2πiα| ≤ 1

2α....

We turn to major arcs. Let

α =a

q+ β

with β very small, and getting smaller as a function of B. Here we willassume that |β| ≤ B−d+δ′ , for some small δ′ > 0. We put x = qy + z,with z the corresponding residue class modulo q, and obtain

S(α) =∑

x∈Zn+1, ‖x‖≤B

e2πiaqf(x)e2πiβf(x)

=∑‖x‖≤B

e2πiaqf(qy+z)e2πiβf(x)

=∑z

e2πiaqf(z)(

∑‖y‖≤B/q

e2πiβf(x))

=∑z

e2πiaqf(z)

∫‖y‖≤B/q

e2πiβf(x)dy

=∑z

e2πiaqf(z)

qn+1

∫‖x‖≤B

e2πiβf(x)dx,

where dy = qn+1dx. The passage∑7→∫

is justified for our choice ofsmall β - the difference will be adsorbed in the error term in (4.2). Wehave obtained∫ 1

0

S(α)dα ∼∑a,q

∑z

e2πiaqf(z)

qn+1·∫|β|≤B−d+δ

∫‖x‖≤B

e2πiβf(x)dxdβ.

We first deal with the integral on the right, called the singular integral.Put β′ = βBd and x′ = x/B. The change of variables leads to∫|β|≤ 1

Bd−δ

dβ

∫‖x‖≤Be2πiβBdf( x

B)Bn+1d(

x

B) = Bn+1−d

∫|β′|≤Bδ

∫‖x′‖≤1

e2πiβ′f(x′)d(x′).

We see the appearance of the main term Bn−d and the density

τ∞ :=

∫ 1

0

dβ′∫‖x‖≤1

e2πiβ′f(x′)dx′.

44 YURI TSCHINKEL

Now we analyze the singular integral

σQ :=∑a,q

∑z

e2πiaqf(z)

qn,

where the outer sum runs over positive coprime integers a, q, a < q andq < Q, and the inner sum over residue classes z ∈ (Z/q)n+1. This sumhas the following properties

(1) multiplicativity in q, in particular we have

σ :=∏p

(∞∑i=0

A(pi)).

with σQ → σ, for Q→∞, (with small error term),(2)

k∑i=0

A(pi)

pi(n+1)=%(f, pk)

pkn,

where

%(f, pk) := # z mod pk | f(z) = 0 mod pk .

Here, a discrete version of equation (4.3) comes into play:

#solutions mod pk =1

pk

pk−1∑a=0

∑z

e2πi

af(z)

pk

However, our sums run over a with (a, p) = 1. A rearranging of termsleads to

%(f, pk)

pkn=

k∑i=0

∑(a,pk)=pi

∑z

1

pk(n+1)e

2πi apkf(z)

=k∑i=0

∑( api,pk−i)=1

∑z

1

pk(n+1)e

2πia/pi

pk−if(z) · p(n+1)i

=k∑i=0

1

p(n+1)(k−i)

∑(a,pk−i)=1

∑z

e2πi a

pk−if(z)

=k∑i=0

1

p(n+1)(k−i) · A(pk−i).


In conclusion,

eqn:eulereqn:euler (4.6) σ =∏p

τp, where τp = limk→∞

%(f, pk)

pnk.

As soon as there is at least one (nonsingular) solution f(z) = 0 mod p,τp 6= 0, and in fact, for almost all p,

%(f, pk)

pnk=%(f, p)

pn,

by Hensel’s lemma. Moreover, if τp 6= 0 for all p, the the Euler productin equation (4.6) converges.

Let us illustrate this in the example of Fermat type equations

f(x) = a0xd + · · ·+ anx

dn = 0.

Using properties of Jacobi sums one can show that

%(f, p) = pn + E, with E = O((p− 1)p(n−1)/2),

so that ∣∣∣∣%(f, p)

pn−1− 1

∣∣∣∣ ≤ C

p(n+1)/2.

The corresponding Euler product∏p

%(f, p)

pn∏p

(1 +C

p(n+1)/2)

is convergent.Some historical background: the cicle method was firmly established

in the series of papers of Hardy and Littlewood Partitio numerum.They comment: “A method of great power and wide scope, applicableto almost any problem concernign the decomposition of integers intoparts of a particular kind, and to many against which it is difficult tosuggest any other obvious method of attack.”

chap:funct4.9. Function fields: heuristics. Let p be a prime and put q = pn.Let Λ be a convex n-dimensional cone in Rn with vertex at 0. Let

f1, f2 : Rn → R

be two linear functions such that

• fi(Zn) ⊂ Z;• f2(x) > 0 for all x ∈ Λ \ 0;• there exists an x ∈ Λ \ 0 such that f(x) > 0.

46 YURI TSCHINKEL

Put

M = ∪λMλ, λ ∈ Zn ∩ Λ

and

|Mλ| := qmax(0,f1(λ)), ϕ(m) := qf2(λ), for m ∈Mλ.

Then the series

F (s) =∑

λ∈Λ∩Zn

|Mλ|qsf2(λ)

converges for

<(s) > a := maxx∈Λ

(f1(x)/f2(x)) > 0.

What happens around s = a? Choose an ε > 0 and decompose thecone

Λ := Λ+ε ∪ Λ−ε ,

whereΛ+ε := x ∈ Λ | f1(x)/f2(x) ≥ a− ε

Λ−ε := x ∈ Λ | f1(x)/f2(x) < a− εTherefore,

F (s) = F+ε + F−ε ,

where F−ε converges absolutely for <(s) > a− ε.Now we make some assumptions concerning Λ: suppose that for all

ε ∈ Q>0, the cone Λ+ε is a rational finitely generated polyhedral cone.

Then

Λaε := x | f1(x)/f2(x) = a

is a face of Λ+ε , and thus also finitely generated polyhedral.

Lemma 4.9.1. There exists a function Gε(s), holomorphic for <(s) >a− ε, such that

Fε(s) =Gε(s)

(s− a)b,

where b is the dimension of the face Λaε .

Proof. For y ∈ Q>0 we put

P (y) := x |x ∈ Λ, f2(x) = y

Consider the expansion

F (s) =∑y∈N

∑λ∈P (y)∩Zn

qf1(λ)−sf2(λ)


Replacing by the integral, we obtain (with w = yz)

=

∫ ∞0

dy

(∫P (y)

qf1(w)−sf2(w)dw

)=

∫ ∞0

dy

(∫P (1)

yn−1qf1(w)−sf2(y)dz

)=

∫P (1)

dz

∫ ∞0

yn−1q(f1(w)−s)ydy

=

∫P (1)

dz1

(s− f1(z))n

∫ ∞0

un−1q−udu

=Γ(n)

(log(q))n

∫P (1)

1

(s− f1(z))ndz.

(4.7)

It is already clear that we get a singularity at s = max(f1(z)) on P (1),which is a. In general, let f be a linear function and

Φ(s) :=

∫∆

(s− f(x))−ndΩ

where ∆ is a polytope of dimension n−1. Then Φ is a rational functionin s, with an asymptotic at s = a given by

volf,a(b− 1)!

(n− 1)!(s− a)−b,

where ∆f,a is the polytope ∆ ∩ f(x) = a, volf,a is its volume andb = 1 + dim ∆f,a.

Let C be a curve of genus g over the finite field Fq and F its functionfield. Let X be a variety over Fq of dimension n. Then V := X × S isa variety over F . Every F -rational point x of V gives rise to a sectionx of the map V → S. We have a pairing

A1(V )× An(V )→ Z

between the groups of (numerical) equivalence classes of codimension1-cycles and codimension n-cycles. We have

An(V ) = An(X)⊗ A1(S)⊕ An−1(X)⊗ A0(S)

andA1 = A1(X)⊕ ZL = (LX , `)−KV = (−KX , 2− 2g)

48 YURI TSCHINKEL

Let L be a very ample line bundle on V . Then

q(L,x)

is the height of the point x with respect to L. The height zeta functiontakes the form

Z(s) =∑

x∈V (F )

q−(L,x)s

=∑

y∈An(X)

N(q)a−[(LX ,y)+`]s,

where

N(q) := #x ∈ V (F ) | cl(x) = y.

We proceed to give some heuristic(!) bound on N(q). The cycles in agiven class y are parametrized by an algebraic variety My and

dim My(x) ≥ χ(NV |x)

(the Euler characteristic). More precisely, the local ring on the modulispace is the quotient of a powerseries ring with h0(NV |x) generators byh1(NV |x) relations. Our main heuristic assumption is that

N(q) ∼ qdim My ∼ qχ(NV |x).

This assumption fails, for example, for points contained in “excep-tional” (accumulating) subvarieties.

By the short exact sequence

0→ Tx → TV |x → NV |x → 0

we have

χ(TV |x) = (−KV , x) + (n+ 1)χ(Ox)χ(NV |x) = (−KX , cl(x)) + nχ(Ox)

From now on we consider a modified height zeta function

Zmod(s) :=∑

qχ(NV |x)−(L,x)s.

We observe that its analytic properties are determined by the ratiobetween two linear functions

(−KX , ·) and (L, ·).


The relevant cone Λ is the cone spanned by classes of (maximally mov-ing) effective curves. The finite generation of this cone for Fano vari-eties is one of the main results of Mori’s theory. We conclude that

NU(L,B) ∼ Ba(log(B))b−1,

wherea = a(L) = max

z∈Λ((−KX , z)/(L, z))

and b = b(L) is the dimension of the face of the cone where this maxi-mum is achieved.

sect:metrization4.10. Metrizations of line bundles. In this section we discuss arefined theory of height functions, based on the notion of an adelicallymetrized line bundle.

Let F be a number field and disc(F ) the discriminant of F (over Q).The set of places of F will be denoted by Val(F ). We shall write v|∞if v is archimedean and v - ∞ if v is nonarchimedean. For any placev of F we denote by Fv the completion of F at v and by ov the ringof v-adic integers (for v - ∞). Let qv be the cardinality of the residuefield Fv of Fv for nonarchimedean valuations. The local absolute value| · |v on Fv is the multiplier of the Haar measure, i.e., d(axv) = |a|vdxvfor some Haar measure dxv on Fv. We denote by A = AF =

∏′v Fv the

adele ring of F . We have the product formula∏v∈Val(F )

|a|v = 1, for all a ∈ F ∗.

defn:metri Definition 4.10.1. Let X be an algebraic variety over F and L a linebundle on X. A v-adic metric on L is a family (‖ · ‖x)x∈X(Fv) of v-adicBanach norms on the fibers Lx such that for all Zariski open subsetsU ⊂ X and every section f ∈ H0(U,L) the map

U(Fv)→ R, x 7→ ‖f‖x,is continuous in the v-adic topology on U(Fv).

exam:metr Example 4.10.2. Assume that L is generated by global sections. Choosea basis (fj)j∈[0,...,n] of H0(X,L) (over F ). If f is a section such thatf(x) 6= 0 then define

‖f‖x := max0≤j≤n

(| fjf

(x)|v)−1,

otherwise ‖0‖x := 0. This defines a v-adic metric on L. Of course, thismetric depends on the choice of (fj)j∈[0,...,n].

50 YURI TSCHINKEL

defn:ad-metric Definition 4.10.3. Assume that L is generated by global sections. Anadelic metric on L is a collection of v-adic metrics, for every v ∈ Val(F ),such that for all but finitely many v ∈ Val(F ) the v-adic metric on Lis defined by means of some fixed basis (fj)j∈[0,...,n] of H0(X,L).

We shall write ‖ · ‖A := (‖ · ‖v) for an adelic metric on L and calla pair L = (L, ‖ · ‖A) an adelically metrized line bundle. Metrizationsextend naturally to tensor products and duals of metrized line bundles,which allows to define adelic metrizations on arbitrary line bundles L(on projective X): represent L as L = L1⊗L−1

2 with very ample L1 andL2. Assume that L1, L2 are adelically metrized. An adelic metrizationof L is any metrization which for all but finitely many v is inducedfrom the metrizations on L1, L2.

defn:height Definition 4.10.4. Let L = (L, ‖ · ‖A) be an adelically metrized linebundle on X and f an F -rational section of L. Let U ⊂ X be themaximal Zariski open subset of X where f is defined and does notvanish. For all x = (xv)v ∈ U(A) we define the local

HL,f,v(xv) := ‖f‖−1xv

and the global height function

HL(x) :=∏

v∈Val(F )

HL,f,v(xv).

By the product formula, the restriction of the global height to U(F )does not depend on the choice of f.

exam:p1-height Example 4.10.5. For X = P1 = (x0 : x1) one has Pic(X) = Z, spannedby the class L = [(1 : 0)]. For all f = x0/x1 ∈ Ga(A) we define

HL,f,v(xv) = max(1, |f|v).The restriction of HL =

∏vHL,f,v to Ga(F ) ⊂ P1 is the usual height

on P1 (with respect to the usual metrization of L = O(1)).

exam:unipotent-height Example 4.10.6. Let X be an equivariant compactification of a unipo-tent group G and L a very ample line bundle on X. The spaceH0(X,L), a representation space for G, has a unique G-invariant sec-tion f, modulo scalars. Indeed, if we had two nonproportional sections,their quotient would be a character of G, which are trivial.

Fix such a section. We have f(gv) 6= 0, for all gv ∈ G(Fv). Put

HL,f,v(gv) = ‖f(gv)‖−1 and HL,f =∏v

HL,f,v.


By the product formula, the global height is independent of the choiceof f.

sect:hei-z4.11. Height zeta functions. Let X be an algebraic variety over aglobal field F , L = (L, ‖ · ‖A) an adelically metrized ample invertiblesheaf on X, HL a height function associated to L, U a subvariety of X,aU(L) the convergence abscissa of the height zeta function

ZU(L, s) :=∑

x∈U(F )

HL(x)−s.

prop:al Proposition 4.11.1.

(1) The value of aU(L) depends only on the class of L in NS(X).(2) Either 0 ≤ aU(L) < ∞, or aU(L) = −∞, the latter possibility

corresponding to the case of finite U(F ). If aU(L) > 0 for oneample L then this is so for every ample L.

(3) aU(Lm) = 1maU(L). In general, aU(L) extends uniquely to a

continuous function on Λnef(X), which is inverse linear oneach half-line unless it identically vanishes.

Proof. All statements follow directly from the listed properties of heights(see [?]). In particular,

aU(L) ≤ a(Pn(F ),O(m)) =n+ 1

mfor some n,m. If ZU(L, s) converges at some negative s, then it mustbe a finite sum. Since for two ample heights H,H′ we have

cHm < H′ < c′Hn, c, c′,m, n > 0,

the value of a can only be simultaneously positive or zero. Finally, if Land L′ are close in the (real) topology of NS(V )R, then L−L′ is a linearcombination of ample classes with small coefficients, and so aU(L) isclose to aU(L′).

nota:al Notation 4.11.2. By Property (1) of Proposition 4.11.1, we may writeaU(L) = aU(L).

exam:ab-v Example 4.11.3. For an abelian variety X and ample invertible sheafL we have

HL = exp(q(x) + `(x) +O(1)),

where q is a positive definite quadratic form on the space X(F ) ⊗ Qand ` is a linear form. It follows that aX(L) = 0, although X(F ) maywell be Zariski dense in V . Also

NX(L, B) ∼ log(B)r/2

52 YURI TSCHINKEL

where r = rkX(F ). Hence for a = 0, the power of log(B) in principlecannot be calculated geometrically: it depends on the arithmetic of Xand F . The hope is that for a > 0 the situation is more stable.

defn:lin-gr Definition 4.11.4. The arithmetic hypersurface of linear growth ΣU

is

ΣarithU := L ∈ NS(X)R | aU(L) = 1.

Obviously, it defines a completely. More precisely,

prop:alpha Proposition 4.11.5.

• If aU(L) > 0 for some L, then ΣU is nonempty and intersectseach half-line in Λeff(X) in exactly one point.• Σ<

U := L | aU(L) < 1 is convex.

Proof. The first statement is clear. The second follows from the Holderinequality: if

0 < σ, σ′ ≤ 1 and σ + σ′ = 1

then

H−σL (x)Hσ′

L (x) ≤ σHL(x)−1 + σ′HL(x)−1

so that from L,L′ ∈ Σ<U it follows that σL+ σ′L′ ∈ Σ<

U .

When rk NS(X) = 1, ΣU is either empty, or consists of one point.Schanuel’s theorem ?? implies that for Pn(F ), this point is the anti-canonical class.

defn:accu-sub Definition 4.11.6. A subvariety X ⊂ U ⊂ V is called point accumu-lating, or simply accumulating (in U with respect to L), if

aU(L) = aX(L) > aU\X(L).

It is called weakly accumulating, if

aU(L) = aX(L) = aU\X(L).

exem:accum Example 4.11.7. If we blow up an F -point of an abelian variety X, theexceptional divisor will be an accumulating subvariety in the resultingvariety, although to prove this we must analyze the height with respectto the exceptional divisor, which is not quite obvious.

We will see in Theorem ?? that for the product Pn1 × · · · × Pnk =X, with nj > 0, every fiber of a partial projection will be weaklyaccumulating with respect to the anticanonical class.


The role of accumulating subvarieties in our subsequent analysis willbe different for various classes of varieties, but we shall generally try topinpoint them in a geometric way. For example, on Fano varieties weshall have to remove the −KX-accumulating subvarieties if we want toensure the linear growth conjecture.

The role of weakly accumulating subvarieties is that they open pos-sibilities to obtain exact lower bounds for the power growth rate ofX(F ) by analyzing subvarieties of smaller dimension.

sect:manin4.12. Manin’s conjecture. The following picture emerged from theanalysis of examples such as Pn, flag varieties, complete intersectionsof small degree [?], [BM90].

Let X be a smooth projective variety with ample anticanonical classover a number field F0. The conjectures below describe the asymptoticof rational points of bounded height in a stable situation, i.e., aftera sufficiently large finite extension F/F0 and passing to a sufficientlysmall Zariski dense subset X ⊂ X.

conj:linear Conjecture 4.12.1 (Linear growth conjecture). One has

eqn:manineqn:manin (4.8) B1 N(X(F ),−KX) B1+ε.

conj:power Conjecture 4.12.2 (The power of log).

eqn:logeqn:log (4.9) N(X(F ),−KX) ∼ B1 log(B)r−1,

where r = rk Pic(XF ).

Conjecture 4.12.3 (General polarizations / linear growth). Everysmooth projective X with −KX ∈ Λbig(X) has a dense Zariski opensubset X such that

ΣarithX = Σgeom

X ,

(see Definitions 4.11.4, ??.

The next level of precision requires that Λeff(X) is a finitely generatedpolyhedral cone. By Theorem 1.1.5, this holds when X is Fano.

Conjecture 4.12.4 (General polarizations / power of log). Let L bean ample line bundle and a(L), b(L) the constants defined in ... Then

eqn:gen-poleqn:gen-pol (4.10) N(X(F ), L) ∼ Ba(L) log(B)b(L)−1.

54 YURI TSCHINKEL

sect:counter4.13. Counterexamples. At present, no counterexamples to Conjec-ture 4.12.1 are known. However, Conjecture 4.12.2 fails in dimension3. The geometric reason for this failure comes from Mori fiber spaces,more specifially from “unexpected” jumps in the rank of the Picardgroup in fibrations.

Let X ⊂ Pn be a smooth hypersurface. We know, by Lefschetz, thatPic(X) = Pic(Pn) = Z, for n ≥ 4. However, this may fail when X hasdimension 2. Moreover, the variation of the rank of the Picard group ina family of surfaces Xt over a number field F may be nontrivial, evenwhen geometrically, i.e., over the algebraic closure F of F , the rank isconstant.

Concretely, consider a hypersurface X ⊂ P3x× P3

y given by a form ofbidegree (1,3):

3∑j=0

xjy3j = 0.

By Lefschetz, the Picard group Pic(X) = Z2, with the basis of hyper-plane sections of P3

x, resp. P3y, and the anticanonical class is computed

as in Example ??

−KX = (3, 1).

Projection onto P3y exhibits X as a P2-fibration over P3. The second

Mori fiber space structure on X is given by projection to P3x, with fibers

diagonal cubic surfaces. The restriction of −KX to each (smooth) fiberXx is the anticanonical class of the fiber.

By ..., rk Pic(Xx) varies between 1 and 4. For example, if F contains√−3, then the rk Pic(Xx) = 4 whenever all xj are cubes in F . The

lower bounds in Section ?? show that

NXx(−KXx ,B) ∼ B log(B)3

for all such fibers, all dense Zariski open subsets Xx and all F . On theother hand, Conjecture ?? implies that

NX(−KX , B) ∼ B log(B),

for some Zariski open X ⊂ X, over a sufficiently large number field F .However, every Zariski open subset X ⊂ X intersects infinitely manyfibers Xx with rk Pic(Xx) = 4 in a dense Zariski open subset. This isa contradiction.


secct:refine4.14. Peyre’s refinement. The refinement concerns the conjecturedasymptotic formula (4.9). Fix a metrization of −KX . The expectationis that

eqn:peyreeqn:peyre (4.11)N(U(F ), KX) = c(−KX) · B1 log(B)r−1(1 + o(1)), as B→∞,

with r = rk Pic(X). Peyre’s achievement was to give a conceptualinterpretation of the constant c(−KX). Here we explain the key stepsof his construction.

Let F be a number field and Fv its v-adic completion. Let X bea smooth algebraic variety over F of dimension d equipped with anadelically metrized line bundle K = KX = (KX , ‖ · · · ‖A). Fix a pointx ∈ X(Fv) and let x1, . . . , xd be local analytic coordinates in an analyticneighborhood Ux of x giving a homeomorphism

φ : U∼−→ F d

v .

Let dy1 ∧ . . . ∧ dyd be the standard differential form on F dv and f :=

φ∗(dy1 ∧ . . . ∧ dyd) its pullback to U . Note that f is a local section ofthe canonical sheaf KX and that a v-adic metric ‖ · ‖v on KX gives riseto a norm ‖f(u)‖v ∈ R>0, for each u ∈ Ux. Let dµv = dy1 · · · dyd be thestandard Haar measure, normalized by∫

odv

dµv =1

dvd/2,

where dv is the local different (which equals 1 for almost all v).Define the local v-adic measure ωK,v on Ux via∫

W

ωK,v =

∫φ(W )

‖f(φ−1(y))‖vdµv,

for every open W ⊂ Ux. This local measure glues to a measure ωK,v onX(Fv). Indeed, changing the coordinates, ....

Let X be a model of X over the integers oF and let v be a place ofgood reduction. Let Fv = ov/mv be the corresponding finite field andput qv = #Fv. Since X is projective, we have

X(Fv) = X (ov)→ X (Fv).

56 YURI TSCHINKEL

We have∫X(Fv)

ωK,v =∑

xv∈X(Fv)

∫ωK,v

=X(Fv)qdv

= 1 +Trv(H

2d−1et (XFv))√qv

+Trv(H

2d−2et (XFv))

qv+ · · ·+ 1

qdv,

where Trv is the trace of the v-Frobenius on the `-adic cohomology ofX. Trying to integrate the product measure over X(A) is problematic,since the Euler product ∏

v

X(Fv)qdv

diverges. In all examples of interest to us, the cohomology groupH2d−1et (XFv ,Q`) vanishes. For instance, this holds if the anticanonical

class is ample, by ??. Still the product diverges, since the 1/qv termdoes not vanish, for projective X. There is a standard regularizationprocedure: Choose a finite set S ⊂ Val(F ), including all v | ∞ and allplaces of bad reduction. Put

λv =

Lv(1,Pic(XQ)) v /∈ S

1 v ∈ S ,

where Lv(s,Pic(XQ)) is the local factor of the Artin L-function as-sociated to the Galois representation on the geometric Picard group.Define the regularized Tamagawa measure

ωK,v := λ−1v ωK,v.

Write ωK,S :=∏

v ωK,v and define

eqn:tamagawaeqn:tamagawa (4.12) τ(−KX) := L∗S(1,Pic(XQ)) ·∫X(F )

ωK,S,

whereL∗S(1,Pic(XQ)) := lim

s→1(s− 1)rL∗S(s,Pic(XQ))

and r is the rank of Pic(XF ).

exam:group Example 4.14.1. Let G be a linear algebraic group over F . It carriesan F -rational d-form ω, where d = dim(G). This form is unique,modulo multiplication by nonzero constants. Fixing ω, we obtain anisomorphism KX ' OG, the structure sheaf, which carries a naturaladelic metrization (‖ · · · ‖A).


eqn:ceqn:c (4.13) c(−KX) = α(X)β(X)τ(−KX),

where

• α(X) := α(Λeff(X),−KX),• β(X) := Br(X)/Br(F ),

and τ(−KX) is the constant defined in equation 4.12.

• f = f(x0, . . . , xn) ∈ Z[x0, . . . , xn], homogeneous, degree d• X = Xf ⊂ Pn corresponding hypersurface• x ∈ X(Q)⇔ x = (x0 : . . . : xn) ∈ (Qn+1 \ 0) /± with f(x) = 0• choose a representative x ∈

(Zn+1

prim \ 0)/± and put

H(x) := maxj

(|xj|).

• define the counting function

N(X,B) := #x | f(x) = 0, x primitive ,H(x) ≤ Bsect:general-polarize

4.15. General polarizations. I follow closely the exposition in [BT98a].Let E/F be some finite Galois extension such that all of the follow-ing constructions are defined over E. Let (X,L) be a smooth quasi-projective d-dimensional variety together with a metrized very ampleline bundle L which embeds X in some projective space Pn. We de-

note by XL

the normalization of the projective closure of X ⊂ Pn. In

general, XL

is singular. We will introduce several notions relying on aresolution of singularities

ρ : X → XL.

Naturally, the defined objects will be independent of the choice of theresolution.

For Λ ⊂ NS(X)R we define

a(Λ,L) := a(Λ, ρ∗L).

We will always assume that a(Λeff(X),L) > 0.

defn:primitive Definition 4.15.1. A pair (X,L) is called primitive if a(Λeff(X),L) ∈Q>0 and if there exists a resolution of singularities

ρ : X → XL

such that for some k ∈ N((ρ∗L)⊗a(Λeff(X),L) ⊗KX)⊗k = O(D),

where D is a rigid effective divisor (h0(X,O(νD)) = 1 for all ν 0).

58 YURI TSCHINKEL

Example 4.15.2. of a primitive pair: (X,−KX), where X is a smoothprojective Fano variety and −KX is a metrized anticanonical line bun-dle.

Let k ∈ N be such that a(Λ,L)k ∈ N and consider

R(Λ,L) := ⊕ν≥0H0(X, (((ρ∗L)a(Λ,L) ⊗KX)⊗k)⊗ν).

As explained in Section ??, in both cases (Λ = Λample or Λ = Λeff ) it isexpected that R(Λ,L) is finitely generated and that we have a fibration

π = πL : X → Y L,

where Y L = Proj(R(L,Λ)). For Λ = Λeff(X) the generic fiber of π is(expected to be) a primitive variety in the sense of Definition 4.15.1.More precisely, there should be a diagram:

ρ : X → XL ⊃ X

↓Y L

such that:• dim(Y L) < dim(X);• there exists a Zariski open U ⊂ Y L such that for all y ∈ U(C)

the pair (Xy,Ly) is primitive (here Xy = π−1(y) ∩ X and Ly is therestriction of L to Xy);• for all y ∈ U(C) we have a(Λeff(X),L) = a(Λeff(Xy),Ly);• For all k ∈ N such that a(Λeff(X),L)k ∈ N the vector bundle

Lk := R0π∗(((ρ∗L)⊗a(Λeff(X),L) ⊗KX)⊗k)

is in fact an ample invertible sheaf on Y L.Such a fibration will be called an L-primitive fibration. A variety

may admit several primitive fibrations.

Example 4.15.3. Let X ⊂ Pn1 × Pn2 (n ≥ 2) be a hypersurface given bya bi-homogeneous form of bi-degree (d1, d2). Both projections X → Pn1andX → Pn2 are L-primitive, for appropriate L. In particular, for n = 3and (d1, d2) = (1, 3) there are two distinct −KX-primitive fibrations:one onto a point and another onto P3

1.


sect:tama4.16. Tamagawa numbers. For smooth projective Fano varieties Xwith an adelically metrized anticanonical line bundle Peyre defined in[Pey95] a Tamagawa number, generalizing the classical construction forlinear algebraic groups. We need to further generalize this to primitivepairs.

Abbreviate a(L) = a(Λeff(X),L) and let (X,L) be a primitive pairsuch that

O(D) := ((ρ∗L)⊗a(L) ⊗KX)⊗k,

where k is such that a(L)k ∈ N and D is a rigid effective divisor as inDefinition 4.15.1. Choose an F -rational section g ∈ H0(X,O(D)); it isunique up to multiplication by F ∗. Choose local analytic coordinatesx1,v, . . . , xd,v in a neighborhood Ux of x ∈ X(Fv). In Ux the section ghas a representation

g = fka(L)(dx1,v ∧ . . . ∧ dxd,v)k,

where f is a local section of L. This defines a local v-adic measure inUx by

ωL,g,v := ‖f‖a(L)xv dx1,v · · · dxd,v,

where dx1,v · · · dxd,v is the Haar measure on F dv normalized by vol(odv) =

1. A standard argument shows that ωL,g,v glues to a v-adic measure onX(Fv). The restriction of this measure to X(Fv) does not depend on

the choice of the resolution ρ : X → XL. Thus we have a measure on

X(Fv).Denote by (Dj)j∈J the irreducible components of the support of D

and by

Pic(X,L) := Pic(X \ ∪j∈JDj).

The Galois group Γ acts on Pic(X,L). Let S be a finite set of valuationsof bad reduction for the data (ρ,Dj, etc.), including the archimedeanvaluations. Put λv = 1 for v ∈ S, λv = Lv(1,Pic(X,L)) (for v 6∈ S)and

ωL := L∗S(1,Pic(X,L))|disc(F )|−d/2∏v

λ−1v ωL,g,v.

(Here Lv is the local factor of the Artin L-function associated to theΓ-module Pic(X,L) and L∗S(1,Pic(X,L)) is the residue at s = 1 of thepartial Artin L-function.) By the product formula, the measure does

60 YURI TSCHINKEL

not depend on the choice of the F -rational section g. Define

τL(X) :=

∫X(F )

ωL,

where X(F ) ⊂ X(A) is the closure of X(F ) in the direct producttopology. The convergence of the Euler product follows from

h1(X,OX) = h2(X,OX) = 0.

We have a map

ρ : Pic(X)R → Pic(X,L)R

and we denote by

Λeff(X,L) := ρ(Λeff(X)) ⊂ Pic(X,L)R.

Let (A,Λ) be a pair consisting of a lattice and a strictly convex(closed) cone in AR: Λ ∩−Λ = 0. Let (A, Λ) the pair consisting of thedual lattice and the dual cone defined by

Λ := λ ∈ AR | 〈λ′, λ〉 ≥ 0, ∀λ′ ∈ Λ.The lattice A determines the normalization of the Lebesgue measureda on AR (covolume =1). For a ∈ AC define

eqn:gameqn:gam (4.14) XΛ(a) :=

∫Λ

e−〈a,a〉da.

The integral converges absolutely and uniformly for <(a) in compactscontained in the interior Λ of Λ.

defn:chi Definition 4.16.1. Assume that X is smooth, NS(X) = Pic(X) andthat −KX is in the interior of Λeff(X). We define

α(X) := XΛeff(X)(−KX).

Remark 4.16.2. This constant measures the volume of the polytopeobtained by intersecting the affine hyperplane (−KX , ·) = 1 with thedual to the cone of effective divisors Λeff(X) in the dual to the Neron-Severi group. The explicit determination of α(X) can be a seriousproblem. For Del Pezzo surfaces, these volumes are given in Section 1.9.For example, let X be the moduli space M0,6 (see Example ??). Thedual to the cone Λeff(X) has 3905 generators (in a 16-dimensional vectorspace), forming 25 orbits under the action of the symmetric group S6.

defn:tau Definition 4.16.3. Let (X,L) be a primitive pair as above. Define

c(X,L) := XΛeff(X,L)(ρ(−KX)) · |H1(Γ,Pic(X,L))| · τL(X).


Example 4.16.4. Let us return to the Example ??. For the image ofM0,6 under f3 (the Segre cubic) one knows an upper and lower bound:

c′B2 log(B)5 ≤ NU(L3,B) ≤ c′′B2 log(B)5,

for an appropriate Zariski open U and some constants c′, c′′ ≥ 0 [VW95].An asymptotic formula for NU(L3,B) of this shape would be compati-ble with the description in Theorem ??. Moreover, we are now in theposition to specify a constant c which should appear in this asymptotic(answering a question in [VW95]). Indeed, the Segre cubic threefoldis singular (it has 10 isolated double points). The blow-up X = S3 of5 points on P3 is a desingularization of S3. Its Picard group Pic(X)is freely generated by the classes L,Ej (for j = 1, . . . , 5). The effec-tive cone is generated by the classes Ej, L − (Ei + Ej + Ek) and theanticanonical class is given by

−KX = 4L− 2(E1 + · · ·+ E5).

The line bundle on S3 giving the Segre embedding pulls back to L =−1/2 · KX . Clearly, a(L) = 2 and b(L) = rk Pic(X) = 6 (see also[BT98b], Section 5.2). The predicted leading constant c = α · τ , where

α =25

5!XΛ(−KX)

andτ = τ∞ ·

∏p

(1− 1/p)6(1 + 6/p+ 6/p2 + 1/p3),

(τ∞ is the “singular integral” - the archimedean density of X).

If (X,L) is not primitive then, by Section ??, some Zariski opensubset U ⊂ X admits a primitive fibration: there is a diagram

X → XL

↓Y L

such that for all y ∈ Y L(F ) the pair (Uy,Ly) is primitive. Then

c(U,L) :=∑y∈Y 0

c(Uy,Ly),

where the right side is a (possibly infinite converging (!)) sum over thesubset Y 0 ⊂ Y L(F ) of all those fibers Uy where

a(L) = a(Ly) and rk Pic(X,L)Ga = rk Pic(Xy,Ly)Ga .

62 YURI TSCHINKEL

In Section 4.1 we will see that even if we start with pairs (X,L)where X is a smooth projective variety and L is a very ample adelicallymetrized line bundle on X we still need to consider singular varieties.

sect:tam-height4.17. Tamagawa number as a height. For a ∈ P(Q), let Sa ⊂ P3

be the diagonal cubic surface fibration

eqn:aaeqn:aa (4.15) a0x30 + a1x

31 + a2x

32 + a3x

33 = 0,

considered in Section 4.13. Let H : P3(Q) → R>0 be the standardheight as in Section 4.1.

Theorem 4.17.1. [EJ] For all ε > 0 there exists a constant c = c(ε)exam:cub-tamsuch that

1

τ(Sa)≥ c H

(1

a0

: . . . :1

a3

)1/3−ε

In particular, we have the following fundamental finiteness property:for B > 0 there are finitely many a ∈ P3(Q) such that τ(Sa) > B.

A similar, but unconditional, result holds for 3 dimensional cubicsand quartics

Theorem 4.17.2.sect:smallest

4.18. Smallest points. [Sie69]For a discussion of bounds of diophantine equations in terms of the

height of the equation, see [Mas02].[NP89]A sample result in this direction is [Pit71], [NP89]: Let

eqn:diageqn:diag (4.16)n∑i=0

aixdi = 0,

with d odd and let a = (a0 ldots, an) ∈ Zn+1 be a vector with nonzerocoordinates. For n d (e.g., n = 2d + 1) and any ε > 0 there exists aconstant c such that (4.16) has a solution x with

n∑i=0

|aixdi | < c∏|ai|d+ε.

For d ≥ 12, one can work with n ∼ 4d2 log(d). There have been aseveral improvements of this result for specific values of d, e.g. [Cas55],[Die03] for quadrics and [Bak89], [Bru94] for d = 3.


Theorem 4.18.1. [EJ] Let Sa ⊂ P3 be the cubic surface given bythm:smallest-height

ax30 + 4x3

1 + 2x32 + x3

3 = 0.

Assume the Generalized Riemann Hypothesis. Then there does not exista constant c > 0 such that

min H(x) |x ∈ Sa(Q) ≤ c

τ(Sa),

for all a ∈ Z.

5. Counting points via universal torsorssect:count-torsors

5.1. The formalism. [Pey04], [Sal98]sect:toric-torsors

5.2. Toric Del Pezzo surfaces.

Example 5.2.1. Let X = BlY (P 2) be the blowup of the projective planein the subscheme

Y := (1 : 0 : 0) ∪ (0 : 1 : 0) ∪ (0 : 0 : 1),

a toric Del Pezzo surface of degree 6. We can realize it as a subvarietyX ⊂ P1

x × P1y × P1

z given by x0y0z0 = x1y1z1. The anticanonical heightis given by

max(|x0|, |x1|)×max(|y0|, |y1|)×max(|z0|, |z1|).There are six exceptional curves: the preimages of the 3 points and thepreimages of lines joining two of these points.

xyz = t3 ... x2yz = t4

5.3. Torsors over Del Pezzo surfaces.

Example 5.3.1. Quintic DP over Q and Grassmannian. [dlB02]

sect:quartic-dp Example 5.3.2. A quartic Del Pezzo surface S with two singularities oftype A1 can be realized as a blow-up of the following points

p1 = (0 : 0 : 1)p2 = (1 : 0 : 0)p3 = (0 : 1 : 0)p4 = (1 : 0 : 1)p5 = (0 : 1 : 1)

in P2 = (x0 : x1 : x2). The anticanonical line bundle embedds S intoP4:

(x20x1 : x0x

21 : x0x1x2 : x0x2(x0 + x1 − x2) : x1x2(x0 + x1 − x2)).

64 YURI TSCHINKEL

The Picard group is spanned by

Pic(S) = 〈L,E1, · · · , E5〉and Λeff(S) by

E1, · · · , E5

L− E2 − E3, L− E3 − E4, L− E4 − E5, L− E2 − E5

L− E1 − E3 − E5, L− E1 − E2− E4.

The universal torsor is given by the following equations

(23)(3)− (1)(124)(4) + (25)(5) = 0(23)(2)− (1)(135)(5) + (34)(4) = 0(124)(1)(2)− (34)(3) + (45)(5) = 0(25)(2)− (135)(1)(3) + (45)(4) = 0

(23)(45) + (34)(25)− (1)2(124)(135) = 0.

(with variables labeled by the corresponding exceptional curves). In-troducing additional variables

(24)′ := (1)(124), (35)′ := (1)(135)

we see that the above equations define a P1-bundle over a codimensionone subvariety of the Grassmannian Gr(2, 5).

We need to estimate the number of 11-tuples of nonzero integers,satisfying the equations above and subject to the inequalities:

|(135)(124)(23)(1)(2)(3)| ≤ B|(135)(124)(34)(1)(3)(4)| ≤ B

· · ·By symmetry, we can assume that |(2)| ≥ |(4)| and write (2) = (2)′(4)+r2. Now we weaken the first inequality to

|(135)(124)(23)(1)(4)(2)′(3)| ≤ B.

There are O(B log(B)6) 7-tuples of integers satisfying this inequality.

Step 1. Use equation (23)(3) − (1)(124)(4) + (25)(5) to reconstruct(25), (5) with ambiguity O(log(B)).

Step 2. Use (25)(2) − (135)(1)(3) + (45)(4) = 0 to reconstruct theresidue r2 modulo (4). Notice that (25) and (4) are “almost” coprimesince the corresponding exceptional curves are disjoint.

Step 3. Reconstruct (2) and (45).


Step 4. Use (23)(2)− (1)(135)(5) + (34)(4) to reconstruct (34).

In conclusion, if U ⊂ S is the complement to exceptional curves then

NU(−KS,B) = O(B log(B)7).

As explained in Remark ??, we expect that

NU(−KS,B) = B log(B)5(1 + o(1))

as B →∞, where is the constant defined in Chapter ??.

exem:dp4-tors Example 5.3.3. The universal torsor of a smooth quartic Del Pezzosurface, given as a blow up of the five points

p1 = (1 : 0 : 0)p2 = (0 : 1 : 0)p3 = (0 : 0 : 1)p4 = (1 : 1 : 1)p5 = (1 : a2 : a3),

assumed to be in general position, is given by the vanishing of

(14)(23) + (12)(34) − (13)(24) (00)(05)− (12)(34) + (13)(24)− (14)(23)

(00)(05) + a3(a2 − 1)(12)(34) − a2(a3 − 1)(13)(24)

(23)(03) + (24)(04) − (12)(01) (12)(01)− (23)(03) + (24)(04)− (25)(05)

a2(23)(03) + (25)(05) − (12)(01)

(12)(35) − (13)(25) + (15)(23) (00)(04)− (12)(35) + (13)(25)− (15)(23)

(a2 − 1)(12)(35) + (00)(04) − (a3 − 1)(13)(25)

(12)(45) + (14)(25) − (15)(24) (00)(03)− (12)(45) + (14)(25)− (15)(24)(00)(03) + a3(14)(25) − (15)(24)

(13)(45) + (14)(35) − (15)(34) (00)(02)− (13)(45) + (14)(35)− (15)(34)(00)(02) + a2(14)(35) − (15)(34)

(23)(45) + (24)(35) − (25)(34) (00)(01)− (23)(45) + (24)(35)− 25)(34)(00)(01) + a2(24)(35) − a3(25)(34)

(04)(34) + (02)(23) − (01)(13) (13)(01)− (23)(02) + (34)(04) + 35)(05)(05)(35) + a3(02)(23) − (01)(13)

(a2 − 1)(03)(34) + (05)(45) − (a3 − 1)(02)(24) (14)(01)− (24)(02) + (34)(03)− (45)(05)

(03)(34) + (01)(14) − (02)(24)

(04)(14) + (03)(13) − (02)(12) (12)(02)− (13)(03) + (14)(04)− (15)(05)

(05)(15) + a2(03)(13) − a3(02)(12)

a3(02)(25) − a2(03)(35) − (01)(15) (15)(01)− (25)(02) + (35)(03)− (45)(04)

(a3 − 1)(02)(25) − (a2 − 1)(03)(35) − (04)(45)

Connection to the D5-Grassmannian

66 YURI TSCHINKEL

exam:cay-tors Example 5.3.4. The Cayley cubic X3 is the unique cubic hypersurfacein P3 with 4 double points (A1-singularities), the maximal number ofdouble points on a cubic surface. It can be given by the equation

y0y1y2 + y0y1y3 + y0y2y3 + y1y2y3 = 0.

The double points correspond to

(1 : 0 : 0 : 0), (0 : 1 : 0 : 0), (0 : 0 : 1 : 0), (0 : 0 : 0 : 1).

It can be realized as the blow-up of P2 = (x1 : x2 : x3) in the points

q1 = (1 : 0 : 0), q2 = (0 : 1 : 0), q3 = (0 : 0 : 1), q4 = (1 : −1 : 0), q5 = (1 : 0 : −1), q6 = (0 : 1 : −1)

The points lie on a rigid configuration of 7 lines

x1 = 0 (12)(13)(14)(1)x2 = 0 (12)(23)(24)(2)

x3 = 0 (13)(23)(34)(3)

x4 = x1 + x2 + x3 (14)(24)(34)(4)x1 + x3 = 0 (13)(24)(13, 24)

x2 + x3 = 0 (23)(14)(14, 23)

x1 + x2 = 0 (12)(34)(12, 34).

The proper transform of the line xj is the (−2)-curve correspondingto (j). The curves corresponding to (ij), (ij, kl) are (−1)-curves. Theaccumulating subvarieties are exceptional curves. The (anticanonical)embeddding X3 → P3 is given by the linear system:

s1 = x1x2x3

s2 = x2x3x4

s3 = x1x3x4

s4 = x1x2x4

The counting problem is: estimate

N(B) = #(x1, x2, x3) ∈ Z3prim/±, | max i(|si|)/ gcd(si) ≤ B,

where the triple xj is subject to the conditions

xi 6= 0, (i = 1, . . . , 4) xj + xi 6= 0(1 ≤ i < j ≤ 3).

We expect ∼ B log(B)6 solutions. After dividing by the coordinates bytheir gcd, we obtain

s′1 = (1)(2)(3)(12)(13)(23)s′2 = (2)(3)(4)(23)(24)(34)s′3 = (1)(3)(4)(13)(14)(34)s′4 = (1)(2)(4)(12)(14)(24)

(These are special sections in the anticanonical series, other decompos-able sections are: (1)(2)(12)2(12, 34) and (12, 34)(13, 24)(14, 23), for


example.) The conic bundles on X3 produce the following equationsfor the universal torsor:

I (1)(13)(14) + (2)(23)(24) = (34)(12, 34)II (1)(12)(14) + (3)(23)(34) = (24)(13, 24)

III (2)(12)(24) + (3)(13)(34) = (14)(14, 23)

IV −(3)(13)(23) + (4)(14)(24) = (12)(12, 34)V −(2)(12)(23) + (4)(14)(34) = (13)(13, 24)

VI −(1)(12)(1) + (4)(24)(34) = (23)(14, 23)VII (2)(4)(24)2 + (1)(3)(13)2 = (12, 34)(14, 23)

VIII −(1)(2)(12)2 + (3)(4)(34)2 = (13, 24)(14, 23)

IX (1)(4)(14)2 − (2)(3)(23)2 = (12, 34)(13, 24)

The counting problem is to estimate the number of 13-tuples of nonzerointegers, satisfying the equations above and subject to the inequalitymaxi|s′i| ≤ B. Heath-Brown proved in [HB03] that there exists con-stants 0 < c < c′ such that

cB log(B)6 ≤ N(B) ≤ c′B log(B)6.sect:tors-segre

5.4. Torsors over the Segre cubic. In this section we work over Q.The threefold X = M0,6 can be realized as the blow-up of P3 in thepoints

x0 x1 x2 x3

q1 1 0 0 0q2 0 1 0 0q3 0 0 1 0q4 0 0 0 1q5 1 1 1 1

and in the proper transforms of lines joining two of these points. TheSegre cubic is given as the image of X in P4 under the linear system2L− (E1 + · · ·E5) (quadrics passing through the 5 points):

s1 = (x2 − x3)x1

s2 = x3(x0 − x1)s3 = x0(x1 − x3)s4 = (x0 − x1)x2

s5 = (x1 − x2)(x0 − x3)

It can be realized in P5 = (y0 : . . . : y5) as

S3 := 5∑i=0

y3i =

5∑i=0

yi = 0

(exhibiting the S6-symmetry.) It contains 15 planes, given by the S6-orbit of

y0 + y3 = y1 + y4 = y3 + y5 = 0,

68 YURI TSCHINKEL

and 10 singular double points, given by the S6-orbit of

(1 : 1 : 1 : −1 : −1 : −1).

This is the maximal number of nodes on a cubic threefold and S3 is theunique cubic with this property. The hyperplane sections S3∩yi = 0are Clebsch diagonal cubic surfaces (unique cubic surfaces with S5 assymmetry group. The hyperplane sections S3 ∩ yi = 0 are Clebschcubics, a unique cubic surface with S5-symmetry. The hyperplanesections S3 ∩ yi − yj = 0 are Cayley cubic surfaces (see Section ??).The geometry and symmetry of these and similar varieties are describedin detail in [Hun96].

The counting problem on S3 is: find the number N(B) of all 4-tupelsof (x0, x1, x2, x3) ∈ Z4/± such that

• gcd(x0, x1, x2, x3) = 1;• maxj=1,...,5(|sj|)/| gcd(s1, . . . , s5) ≤ B;• xi 6= 0 and xi − xj 6= 0 for all i, j 6= i.

The last condition is excluding rational points contained in accumulat-ing subvarieties (there are B3 rational points on planes P2 ⊂ P4, withrespect to the O(1)-height). The second condition is the bound on theheight.

First we need to determine

a(L) = infa | aL+KX ∈ Λeff(X),

where L is the line bundle giving the map to P4. We claim that a(L) =2. This follows from the fact that∑

i,j

(ij)

is on the boundary of Λeff(X) (where (ij) is the class in Pic(X) of thepreimage in X of the line lij ⊂ P4 through qi, qj.

Therefore, we expect

N(B) = O(B2+ε)

as B → ∞. In fact, it was shown in [BT98a] that b(L) = 6. Conse-quently, one expects

N(B) ∼ cB log(B)5, as B→∞.


Remark 5.4.1. The difficult part is to keep track of gcd(s1, . . . , s5).Indeed, if we knew that this gcd = 1 we could easily prove the boundO(B2+ε) by observing that there are O(B1+ε) pairs of (positive) integers(x2 − x3, x1) (resp. (x0 − x1, x2)) satisfying (x2 − x3)x1 ≤ B (resp.(x0 − x1)x2 ≤ B). Then we could reconstruct the quadruple

(x2 − x3, x1, x0 − x1, x2)

and consequently(x0, x1, x2, x3)

up to O(B2+ε).

Thus it is necessary to introduce gcd between xj, etc. Again, we usethe symbols (i), (ij), (ijk) for variables on the torsor for X correspond-ing to the classes of the preimages of points, lines, planes resp. Oncewe fix a point (x0, x1, x2, x3) ∈ Z4 (such that gcd(x0, x1, x2, x3) = 1),the values of these coordinates over the corresponding point on X canbe expressed as greatest common divisors. For example, we can write

x3 = (123)(12)(13)(23)(1)(2)(3),

a product of integers (neglecting the sign of x3; in the torsor language,we are looking at the orbit of TNS(Z)). Here is a self-explanatory list:

(123) x3 (12) x2, x3

(124) x2 (13) x1, x3

(125) x2 − x3 (14) x1, x2

(134) x1 (15) x1 − x3, x1 − x2

(135) x1 − x3 (23) x3, x0 − x3

(145) x1 − x2 (24) x2, x0

(234) x0 (25) x3 − x2, x0 − x3

(235) x0 − x3 (34) x1, x0

(245) x0 − x2 (35) x1 − x3, x0 − x1

(345) x0 − x1 (45) x1 − x2, x0 − x1.

After dividing sj by the gcd, we get

s′1 = (125)(134)(12)(15)(25)(13)(14)(34)(1)s′2 = (123)(245)(12)(13)(23)(24)(25)(45)(2)s′3 = (234)(135)(23)(24)(34)(13)(15)(35)(3)s′4 = (345)(124)(34)(35)(45)(12)(14)(24)(4)s′5 = (145)(235)(14)(15)(45)(23)(35)(25)(5)

(observe the symmetry with respect to the permutation (12345)). Weclaim that gcd(s′1, . . . , s

′5) = 1. One can check this directly using the

70 YURI TSCHINKEL

definitions of (i), (ij), (ijk)’s as gcd’s. For example, let us check thatnontrivial divisors d 6= 1 of (1) cannot divide any other s′j. Such ad must divide (123) or (12) or (13) (see s′2). Assume it divides (12).Then it doesn’t divide (13), (14) and (15) (the corresponding divisorsare disjoint). Therefore, d divides (135) (by s′3) and (235) (by s′5).Contradiction (indeed, (135) and (235) correspond to disjoint divisors).Assume that d divides (123). Then it has to divide either (13) or (15)(from s′3) and either (12) or (14) (from s′4). Contradiction.

The integers (i), (ij), (ijk) satisfy a system of relations (these areequations for the torsor induced from fibrations of M0,6 over P1):

I x0 x1 x0 − x1

II x0 x2 x0 − x2

III x0 x3 x0 − x3

IV x1 x2 x1 − x2

V x1 x3 x1 − x3

VI x2 x3 x2 − x3

VII x0 − x1 x0 − x2 x1 − x2

VIII x0 − x1 x0 − x3 x1 − x3

IX x1 − x2 x1 − x3 x2 − x3

X x2 − x3 x0 − x3 x0 − x2

which translates toI (234)(23)(24)(2)− (134)(13)(14)(1) = (345)(45)(35)(5)

II (234)(23)(34)(3)− (124)(12)(14)(1) = (245)(25)(45)(5)III (234)(24)(34)(4)− (123)(12)(13)(1) = (235)(25)(35)(5)

IV (134)(13)(34)(3)− (124)(12)(24)(2) = (145)(15)(45)(5)

V (134)(14)(34)(4)− (123)(12)(23)(2) = (135)(15)(35)(5)VI (124)(14)(24)(4)− (123)(13)(23)(3) = (125)(15)(25)(5)

VII (345)(34)(35)(3)− (245)(24)(25)(2) = −(145)(14)(15)(1)

VIII (345)(34)(45)(4)− (235)(23)(25)(2) = −(135)(13)(15)(1)IX (145)(14)(45)(4)− (135)(13)(35)(3) = −(125)(12)(25)(2)

X (125)(12)(15)(1) + (235)(23)(35)(3) = −(245)(24)(45)(4)

The counting problem now becomes: find all 25-tuples of nonzerointegers satisfying the equations I−X and the inequality max(|s′j|) ≤ B.

rema:gr26 Remark 5.4.2. Note the analogy to the the case of M0,5 (the uniquesplit Del Pezzo surface of degree 5): the variety defined by the aboveequations is the Grassmannian Gr(2, 6) (in its Plucker embedding intoP24).

Remark 5.4.3. Vaughan and Wooley show in [VW95] that there existconstants c, c′ > 0 such that

cB log(B)5 ≤ N(B) ≤ c′B log(B)5.


They use a different (an intermediate) torsor overX - the determinantalvariety given by

det(xij)3×3 = 0.

sect:flag-tors5.5. Flag varieties and torsors. Natural examples of torsors undertori arise from flag varieties. Let G be a semi-simple algebraic group,P ⊂ G a parabolic subgroup. The flag variety P\G admits an actionby any subtorus of the maximal torus in G on the right. Choosinga linearization for this action and passing to the quotient we obtaina plethora of examples of nonhomogeneous varieties X whose torsorscarry additional symmetries. These may be helpful in the countingrational points on X. Indeed, we have already seen such examples (DelPezzo surface of degree 5 ?? and the Segre cubic, Section 5.4).

In this section we expand on these examples.

exam:tors-g2 Example 5.5.1. A flag variety for the group G2 is the quadric hypersur-face

v1u1 + v2u2 + v3u3 + z2 = 0,

where the torus G2m ⊂ G2 acts as

vj 7→ λjvj, j = 1, 2 v3 7→ (λ1λ2)−1v3

uj 7→ λ−1j uj, j = 1, 2 u3 7→ λ1λ2u3.

The quotient by G2m is a subvariety in the weighted projective space

P(1, 2, 2, 2, 3, 3) = (z : x1 : x2 : x3 : y1 : y2) with the equations

x0 + x1 + x2 + z2 = 0 and x1x2x3 = y1y2.

6. Height zeta functionssect:zetasect:tools

6.1. Tools from analysis. In this section we collect technical resultsfrom complex and harmonic analysis which will be used in the treat-ment of height zeta functions.

For U ⊂ Rn letTU := s ∈ C | <(s) ∈ U

be the tube domain over U .

thm:convex Theorem 6.1.1 (Convexity principle). Let U ⊂ Rn be a connectedopen subset and U the convex envelope of U , i.e., the smallest con-vex open set containing U . Let Z(s) be a function holomorphic in TU .Then Z(s) is holomorphic in TU .

72 YURI TSCHINKEL

thm:phragmen Theorem 6.1.2 (Phragmen-Lindelof principle). Let φ be a holomor-phic function for <(s) ∈ [σ1, σ2]. Assume that in this domain φ satisfiesthe following bounds

• |φ(s)| = O(eε|t|), for all ε > 0;• |φ(σ1 + it)| = O(|t|k1) and |φ(σs + it) = |O(|t|k2).

Then, for all σ ∈ [σ1, σ2] one has

|φ(σ + it)| = O(|t|k), wherek − k1

σ − σ1

=k2 − k1

σ2 − σ1

.

Using the functional equation and known bounds for Γ(s) in verticalstrips one derives the convexity bound

eqn:bound-zetaeqn:bound-zeta (6.1) |ζ(1

2+ it)| = O(|t|1/4+ε).

More generally, we have the following

prop:phragmen Proposition 6.1.3. Let χ be an unramified character of Gm(AF )/Gm(F ),i.e., χv is trivial on G(ov), for all v - ∞. For all ε > 0 there exists aδ > 0 such that

eqn:l-estimeqn:l-estim (6.2) |L(s, χ)| (1 + |=(χ)|+ |=(s)|)ε, for <(s) > 1− δ.Here =(χ) ∈ ⊕v|∞Gm(Fv)/Gm(ov) ' Rr1+r2−1, with r1, r2 the numberof real, resp. pairs of complex embeddings of F .

thm:tauber Theorem 6.1.4 (Tauberian theorem). Let λn be an increasing se-quences of positive real numbers, with limn→∞ λn = ∞. Let an beanother sequence of positive real numbers and put

Z(s) :=∑n≥1

anλsn.

Assume that this series converges absolutely and uniformly to a homo-morphic function in the tube domain T>a ⊂ C, for some a > 0, andthat it admits a representation

Z(s) =h(s)

(s− a)b,

where h is holomorphic in T>a−ε, for some ε > 0, with h(a) = c > 0,and b ∈ N. Then

N(B) :=∑λn≤B

an ∼c

a(b− 1)!Ba log(B)b−1, for B →∞.

A frequently employed result in analytic number theory is


Theorem 6.1.5 (Poisson formula). Let G be a locally compact abelian

group with Haar measure dg and H ⊂ G a closed subgroup. Let G bethe Pontryagin dual of G, i.e., the group of characters, i.e., continuoushomomorphisms,

χ : G→ S1 ⊂ C∗

into the unit circle. Let f : G→ C be a function, satisfying some mildassumptions (integrability, continuity) and let

f(χ) :=

∫G

f(g) · χ(g)dg

be its Fourier transform. Then there exist Haar measures dh on H anddh⊥ on H⊥, the subgroup of characters trivial on H, such that

eqn:poissoneqn:poisson (6.3)

∫H

fdh =

∫H⊥

fdh⊥.

A standard application is to H = Z ⊂ R = G. In this case H⊥ =H = Z, and the formula reads∑

n∈Z

f(n) =∑n∈Z

f(n).

This is a powerful identity which is used, e.g., to prove the functionalequation and meromorphic continuation of the Riemann zeta function.We will apply Equation (6.3) in the case when G is the group of adelicpoints of an algebraic torus or an additive group, and H is the sub-group of rational points. This will allow us to establish a meromorphiccontinuation of height zeta functions for equivariant compactificationsof these groups.

Another application of the Poisson formula arises as follows: Let Abe a lattice and Λ a convex cone in AR. Let da be the Lebesgue measureon the dual space AR normalized by the dual lattice A). Let

α(Λ, s) :=

∫Λ

e−〈s,a〉da, <(s) ∈ Λ.

be the Laplace transform of the set-theoretic characteristic function ofthe dual cone Λ.

Let π : (A,Λ) → (A, Λ) be a homomorphism, with finite cokernelA′ and kernel B ⊂ A. Normalize db : vol(BR/B) = 1. Then

α(Λ, π(s)) =1

(2π)k−k1

|A′|

∫BR

α(Λ, s + ib)db.

74 YURI TSCHINKEL

In particular,

α(Λ, s) =1

(2π)d

∫MR

n∏j=1

1

(sj + imj)dm.

Oh’s resultssome number theoretic estimates??

sect:setup-groups6.2. Compactifications of groups and homogeneous spaces. Asalready mentioned in Section 3, an easy way to generate examplesof algebraic varieties with many rational points is to use actions ofalgebraic groups. Here we discuss the geometric properties of groupsand their compactifications.

Let G be a linear algebraic group over a field F , and

% : G→ PGLn+1

an algebraic representation over F . Let x ∈ Pn(F ) be a point. Theorbit %(G) · x ⊂ Pn inherits rational points from G(F ). Let H ⊂ G bethe stabilizer of x. In general, we have an exact sequence

1→ H(F )→ G(F )→ G/H(F )→ H1(F,H)→ · · ·

We will only consider examples when (G/H)(F ) = G(F )/H(F ).By construction, the Zariski closure X of %(G)·x is geometrically iso-

morphic to an equivariant compactification of the homogeneous spaceG/H. We have a dictionary

(%, x ∈ Pn)⇔

equivariant compactification X ⊃ G/H,G-linearized very ample line bundle L on X.

Representations of semisimple groups do not deform, and can be char-acterized by combinatorial data: lattices, polytopes etc. Note, how-ever, that the choice of the initial point x ∈ Pn can still give rise tomoduli. On the other hand, the classification of representations ofunipotent groups is a wild problem, already for G = G2

a. In this case,understanding the moduli of representations of a fixed dimension isequivalent to classifying pairs of commuting matrices, up to conjugacy(see [GP69]).

recent Gorodnik/OhBorovoi


sect:principles6.3. Basic principles. Here we explain some common features inthe study of height zeta functions of compactifications of groups andhomogeneous spaces.

In all examples, we have Pic(X) = NS(X), a torsion-free abeliangroup. Choose a basis L1, . . . , Lr of Pic(X) and metrizations Lj =(Lj, ‖ · ‖v). We obtain a height system:

Hj : X(F )→ R>0, for j = 1, . . . , r,

which can be extend to Pic(X)C, by linearity:

eqn:heightseqn:heights (6.4)H : X(F )× Pic(X)C → R>0

(x, s) 7→∏r

j=1 HLj(x)sj ,

where s :=∑r

j=1 sjLj. For each j, the 1-parameter zeta function

ZX(Lj, s) =∑

x∈X(F )

HL(s)−s

converges absolutely to a holomorphic function, for <(s) 0 (seeRemark ??). It follows that

ZX(s) :=∑

x∈X(F )

H(s, x)−1

converges absolutely to a holomorphic function for <(s) contained somecone in Pic(X)R.

Step 1. One introduces a generalized height height pairing

eqn:heights-adeliceqn:heights-adelic (6.5) H =∏v

Hv : G(A)× Pic(X)C → C,

such that the restriction of H to Pic(X) × G(F ) coincides with thepairing in (6.4). Since X is projective, the height zeta function

eqn:zzeqn:zz (6.6) Z(g, s) :=∑

γ∈G(F )

H(γg, s)−1

converges to a function which is continuous in g and holomorphic in sfor <(s) contained in some cone Λ ⊂ Pic(X)R. The standard heightzeta function is obtained by setting g = e, the identity in G(A). Ourgoal is to obtain a meromorphic continuation to the tube domain Tover an open neighborhood of [−KX ] = κ ∈ Pic(X)R and to identifythe poles of Z in this domain.

76 YURI TSCHINKEL

Step 2. It turns out that

Z(g, s) ∈ L2(G(F )\G(A)),

for <(s) 0. This is immediate in the cocompact case, e.g., forG unipotent or semisimple anisotropic, and requires an argument inother cases. The L2-space decomposes into unitary irreducible repre-sentations for the natural action of G(A). We get a formal identity

eqn:formalzeqn:formalz (6.7) Z(g, s) =∑%

Z%(g, s),

where the summation is over all irreducible unitary representations(%,H%) of G(A) occuring in the right regular representation of G(A) inL2(G(F )\G(A)).

Step 3. In many cases, the leading pole of Z(g, s) arises from thetrivial representation, i.e., from the integral

eqn:trivialeqn:trivial (6.8)

∫G(AF )

H(g, s)−1dg =∏v

∫G(Fv)

Hv(gv, s)−1dgv,

where dgv is a Haar measure on G(Fv). To simplify the exposition weassume that

X \G = D = ∪i∈IDi,

where D is a divisor with normal crossings whose components Di aregeometrically irreducible.

We choose integral models for X and Di and observe

G(Fv) ⊂ X(Fv)∼−→ X(Ov)→ X(Fq) = ∪I⊂IDI (Fq),

where

DI := ∩i∈IDi, DI := DI \ ∪I′)IDI′ .

Write s =∑

i siDi and For almost all v, we have:eqn:local-intteqn:local-intt (6.9)∫

G(Fv)

Hv(gv, s)−1dgv = τv(G)−1

(∑I⊂I

#DI (Fq)qdim(X)

∏i∈I

q − 1

qsi−κi+1 − 1

),

where τv(G) is the local Tamagawa number of G and κi is the order ofthe pole of the (unique modulo constants) top-degree differential formon G along Di (SEE ...). The height integrals are geometric versionsof Igusa’s integrals. They are closely related to “motivic” integrals ofBatyrev, Kontsevich, Denef and Loeser (see [?], [?] and [?]).


This allows to regularize explicitely this adelic integral. For example,for unipotent G we have

eqn:global-inteqn:global-int (6.10)

∫G(AF )

H(g, s)−1dg =∏i

ζF (si − κi + 1) · Φ(s),

with Φ(s) holomorphic and absolutely bounded for <(si) > κi − δ, forall i.

Step 4. Next, one has to identify the leading poles of Z%(g, s), and toobtain bounds which are sufficiently uniform in % to yield a meromor-phic continuation of the right side of (6.15). This is nontrivial alreadyfor abelian groups G (see Section ?? for the case when G = Gn

a). More-over, will need to show pointwise convergence of the series, as a functionof g ∈ G(A).

ForG abelian, e.g., an algebraic torus, all unitary representation havedimension one, and equation (6.15) is nothing but the usual Fourierexpansion of a “periodic” function. The adelic Fourier coefficient is anEuler product, and the local integrals can be evaluated explicitly.

For other groups, it is important to have some sort of parametrizationof representations occurring on the right side of the spectral expansion.For example, for unipotent groups such a representation is provided byKirillov’s orbit method (see Section 6.6). For semisimple groups onehas to appeal to Langland’s theory of automorphic representations (seeSection ??).

sect:additive6.4. Additive groups. Let X be an equivariant compactification ofan additive group G = Gn

a . For example, any blowup X = BlY (Pn),with Y ⊂ Pn−1 ⊂ Pn, can be equipped with a structure of an equi-variant compactification of Gn

a . In particular, the Hilbert schemes ofall algebraic subvarieties of Pn−1 appear in the moduli of equivariantcompactifications X as above. Some features of the geometry of suchcompactifications have been explored in [HT99]. The analysis of heightzeta functions has to capture this geometric complexity. In this sectionwe present an approach to height zeta functions developed in [CLT99],[CLT00], and [CLT02].

Indeed, the height pairing (6.5) determines X uniquely ??? (SEEABOVE).

78 YURI TSCHINKEL

The Poisson formula yields

Z(s) =∑

x∈G(F )

H(x, s)−1

=

∫G(AF )

H(x, s)−1dx +∑ψ 6=ψ0

H(ψ, s),

where the sum runs over all nontrivial characters ψ ∈ (G(AF )/G(F ))∗

and

H(ψ, s) =

∫G(AF )

H(x, s)−1ψ(x)dx

is the Fourier transform, with an appropriately normalized Haar mea-sure dx.

exam:p1 Example 6.4.1. The simplest case is G = Ga = A1 ⊂ P1, over F = Q,with the standard height

Hp(x) = max(1, |x|p), H∞(x) =√

1 + x2.

We have

eqn:ga-zetaeqn:ga-zeta (6.11) Z(s) =∑x∈Q

H(x)−s =

∫AQ

H(x)−sdx+∑ψ

H(ψ, s).

The local Haar measure dxp is normalized by vol(Zp) = 1 so that

vol(|x|p = pj) = pj(1− 1

p).

We have∫Qp

Hp(x)−sdxp =

∫Zp

Hp(x)−sdxp +∑j≥1

∫|x|p=pj

Hp(x)−sdxp

= 1 +∑j≥1

p−jsvol(|x|p = pj) =1− p−s

1− p−(s−1)∫R(1 + x2)−s/2dx =

Γ((s− 1)/2)

Γ(s/2).

Now we analyze the contributions from nontrivial characters. Eachsuch character ψ decomposes as a product of local characters, definedas follows:

ψp = ψp,ap : xp 7→ e2πiap·xp , ap ∈ Qp,

ψ∞ = ψ∞,a∞ : x 7→ e2πia∞·x, a ∈ R.


A character is unramified at p if it is trivial on Zp, i.e., ap ∈ Zp.Then ψ = ψa, with a ∈ AQ. A character ψ = ψa is unramified for

all p iff a ∈ Z. Pontryagin duality identifies Qp = Qp, R = R, and(AQ/Q)∗ = Q.

Since Hp is invariant under the translation action by Zp, the local

Fourier transform Hp(ψap) vanishes unless ψp is unramified at p. In par-ticular, only unramified characters are present in the expansion (6.11),i.e., we may assume that ψ = ψa with a ∈ Z \ 0. For p - a, we compute

Hp(s, ψa) = 1 +∑j≥1

p−sj∫|x|p=pj

ψa(xp)dxp = 1− p−s.

Putting together we obtain

Z(s) =ζ(s− 1)

ζ(s)· Γ((s− 1)/2)

Γ(s/2)

+∑a∈Z

∏p-a

1

ζp(s)·∏p|a

Hp(xp)−sdxp ·

∫R(1 + x2)−s/2 · e2πiaxdx

For <(s) > 2− δ, we have the upper bounds

|∏p|a

Hp(xp)−sdxp| |

∏p|a

∫Qp

Hp(xp)−sdxp |a|εeqn:ga-ineqeqn:ga-ineq (6.12)

|∫

R(1 + x2)−s/2 · e2πiaxdx| N

1

(1 + |a|)N, for any N ∈ N,(6.13)

where the second inequality is proved via repeated integration by parts.This gives a meromorphic continuation of Z(s) and its pole at s = 2

(corresponding to −KX = 2L ∈ Z = Pic(P1)). The leading coefficientat this pole is the Tamagawa number defined by Peyre.

Now we turn to the general case.

• Pic(X) = ⊕iZDi

• −KX =∑

i κiDi, with κi ≥ 2• Λeff(X) = ⊕iR≥0Di

Local and global heights are given by

HDi,v(x) := ‖fi(x)‖−1v and HDi(x) =

∏v

HDi,v(x),

where fi is the unique G-invariant section of H0(X,Di). We get a heightpairing:

H : G(AF )× Pic(X)C → C(x,∑

i siDi) 7→∏

i HDi(x)si

80 YURI TSCHINKEL

Similarly, obtain characters of Gna(AF ).

A character is determined by a “linear form” 〈a, ·〉 = fa, on Gna ,

which gives a rational function fa ∈ F (X)∗. We have

div(fa) = Ea −∑i

di(fa)Di

with di ≥ 0, for all i.Define:

• S(a) ⊂ Val(F )• I0(a) := i | di(fa) = 0 ( I

We have

H(ψa, s) =∏

i∈I0(a)

ζF (si − κi + 1) · Φa(s) ·∫G(A∞)

H∞(x, s)−1ψa,∞(x)dx

with Φa(s) holomorphic for <(si) > κi− δ and bounded by (1 + ‖a‖)ε.

Z(s) =∫G(AF )

H(x, s)−1dx+∑I0(I

∑ψa : I0(a)=I H(ψa, s)

=∏

i∈I ζF (si − κi + 1) · Φ(s)

+∑

I⊂I∏

i∈I ζF (si − κi + 1) · ΦI(s)sect:toric

6.5. Toric varieties. Analytic properties of height zeta functions oftoric varieties have been established in [BT95], [BT98a], and [?].

An algebraic torus is a linear algebraic groups T over a field F suchthat

TE ' Gdm,E

for some finite Galois extension E/F . Such an extension is called asplitting field of T . A torus is split if T ' Gd

m,F . The group of algebraiccharacters

M := X∗(T ) = Hom(T, K∗)

is a torsion free Γ := Gal(E/F )-module. The standard notation forits dual, the cocharacters is N := X∗(T ). There is an equivalence ofcategories: d− dimensional integral

Γ− representations,up to equivalence

⇔ d− dimensional

algebraic tori, split over E,up to isomorphism

The local and global theory of tori can be summarized as follows: Thelocal Galois groups Γv := Gal(Ew/Fv) ⊂ Γ act on Mv := MΓv , the


characters of T (Fv). Let Nv be the lattice of local cocharacters. WriteT (ov) ⊂ T (Fv) for a maximal compact subgroup (after choosing anintegral model, it is indeed the group of ov-valued points of T , foralmost all v). Then

T (Fv)/T (ov) → Nv = NΓv ,

an isomorphism for v unramified in E/F . Adelically, we have

T (AF ) ⊃ T 1(AF ) = t |∏v

|m(tv)|v = 1 ∀m ∈MΓ

andT (F ) → T 1(AF ).

Let KT :=∏

v T (ov) be the maximal compact subgroup of T (AF ).

thm:tori-adelic Theorem 6.5.1. We have

• T (AF )/T 1(AF ) = NΓR ;

• T 1(AF )/T (F ) compact;• KT ∩ T (F ) finite;• the map (T (AF )/KT · T (F ))∗ → ⊕v|∞Mv ⊗ R has finite ker-

nel (analogs of roots of 1) and maps the characters to a lattice⊕MΓ

R .

Over algebraically closed fields, complete toric varieties, i.e., equi-variant compactifications of algebraic tori, are described and classifiedby a combinatorial structure (M,N,Σ), where

• M is the free abelian group of finite rank (the algebraic char-acters of the torus T ),• N := Hom(M,Z) is the dual group of cocharacters, and• Σ = σ is a fan, i.e., a collection of strictly convex cones inNR such that(1) 0 ∈ σ for all σ ∈ Σ,(2) NR = ∪σ∈Σσ,(3) every face τ ⊂ σ is in Σ(4) σ ∩ σ′ ∈ Σ and is face of σ, σ′.

A fan Σ is called regular if the generators of every σ ∈ Σ form part of abasis of N . In this case, the corresponding toric variety XΣ is smooth.The toric variety is constructed as follows:

XΣ := ∪σUσ where Uσ := Spec(F [M ∩ σ]),

and σ ⊂ MR is the cone dual to σ ⊂ NR. The fan Σ encodes allgeometric information about XΣ. For example, 1-dimensional genera-tors e1, . . . , en of Σ correspond to boundary divisors D1, . . . , Dn, i.e.,

82 YURI TSCHINKEL

the irreducible components of XΣ \T . There is an explicit criterion forprojectivity and a description of the cohomology ring, cellular structureetc.

Example 6.5.2. The simplest toric variety is X = P1. We have threedistinguished Zariski open subsets:

• P1 ⊃ Gm = Spec(F [x, x−1]) = Spec(F [xZ])• P1 ⊃ A1 = Spec(F [x]) = Spec(F [xZ≥0 ]),• P1 ⊃ A1 = Spec(F [x−1]) = Spec(F [xZ≤0 ])

They correspond to the semi-groups:

• Z dual to 0,• Z≥0 dual to Z≥0,• Z≤0 dual to Z≤0.

Over nonclosed ground fields F one has to account for the action ofthe Galois group of a splitting field E/F . The necessary modificationscan be described as follows. The Galois group Γ acts on M,N,Σ. Afan Σ is called Γ-invariant if γ · σ ∈ Σ, for all γ ∈ Γ, σ ∈ Σ. If Σ is acomplete regular Γ-invariant fan such that, over the splitting field, theresulting toric variety XΣ,E is projective, then it can be descended toa complete algebraic variety XΣ,F over the groundfield F such that

XΣ,E ' XΣ,F ⊗Spec(F ) Spec(E),

as E-varieties with Γ-action.Picard groupThe split case

PL(Σ) - piecewise linear Z-valued functions ϕ on Σdetermined by M ⊃ mσ,ϕσ∈Σ, i.e., by its values sj := ϕ(ej), j =1, . . . , n.

0→M → PL(Σ)π−→ Pic(XΣ)→ 0

• every divisor is equivalent to a linear combination of bound-ary divisors D1, . . . , Dn, and ϕ is determined by its values one1, . . . , en• relations come from characters of T

The nonsplit case

0→MΓ → PL(Σ)Γ π−→ Pic(XΣ)Γ → H1(Γ,M)→ 0


Λeff(XΣ) = π(R≥0D1 + . . .+ R≥0Dn)−KΣ = π(D1 + . . .+Dn)

Example 6.5.3. P1 = x = (x0 : x1) ⊃ Gm

Hv(x) :=

|x0

x1|v if |x0|v ≥ |x1|v

|x1

x0|v otherwise

H(x) :=∏v

Hv(x)

In general, T (Fv)/T (Ov) → Nv. For ϕ ∈ PL(Σ) put

HΣ,v(x, ϕ) := qϕ(xv)v HΣ(x, ϕ) :=

∏v

HΣ,v(x, ϕ),

with qv = e, for v | ∞. This height has the following properties:

• it gives a pairing T (AF )× PL(Σ)C → C;• the restriction to T (F ) × PL(Σ)C descends to a well-defined

pairing

T (F )× Pic(XΣ)C → C;

• T (Ov)-invariance, for all v

Height zeta function:

ZΣ(s) :=∑

x∈T (F )

HΣ(x, ϕs)−1

Poisson formula

ZΣ(s) :=

∫(T (AF )/KT ·T (F ))∗

HΣ(χ, s)dχ

HΣ(χ, s) :=

∫T (AF )

HΣ(x,−ϕs)χ(x)dx

• for χ nontrivial on KT we have HΣ = 0• converges absolutely for <(sj) > 1 (for all j)• Haar measures on T (Fv) normalized by T (Ov)

Example 6.5.4. Consider the projective line P1 over Q. We have

0→M → PL(Σ)→ Pic(P1)→ 0

84 YURI TSCHINKEL

with M = Z and PL(Σ) = Z2 The Fourier transforms of local heightscan be computed as follows:

Hp(χ0, s) = 1 +∑n≥1

p−s1−im +∑n≥1

p−s1+im =ζp(s1 + im)ζp(s2 − im)

ζp(s1 + s2),

H∞(χ0, s) =

∫ ∞0

e(−s1−im)xdx+

∫ ∞0

e(−s2+im)xdx =1

s1 + im+

1

s2 − im.

We obtain

ZP1(s1, s2) =

∫Rζ(s1 + im)ζ(s2 − im) ·

(1

s1 + im+

1

s2 − im

)dm.

The integral converges for <(s1),<(s2) > 1, absolutely and uniformlyon compact subsets. It remains to establish its meromorphically con-tinuation. This can be achieved by shifting the contour of integrationand computing the resulting residues.

It is helpful to compare this approach with the analysis of P1 as anadditive variety in Example ??.

The Fourier transforms of local height functions in the case of Gdm

over Q are given by:

• v -∞:

HΣ,v(χv,−s) =d∑

k=1

∑σ∈Σ(k)

(−1)k∏ej∈σ

1

1− q−(sj+i〈ej ,m〉)v

• v | ∞

HΣ,v(χv,−s) =∑σ∈Σ(d)

∏ej∈σ

1

(sj + i〈ej,m〉)

where χ = χm is the character corresponding to m ∈MR. The generalcase of nonsplit tori over number fields requires more care. We havean exact sequence of Γ-modules:

0→M → PL(Σ)→ Pic(XΣ)→ 0,

with PL(Σ) a permutation module. Duality gives a sequence of groups:

0→ TPic(AF )→ TPL(AF )→ T (AF )

with

TPL(AF ) =k∏j=1

RFj/FGm(AF ) (restriction of scalars)


We get a map

(T (AF )/KT · T (F ))∗ →∏k

j=1

(Gm(AFj)/Gm(Fj)

)∗χ 7→ (χ1, . . . , χk)

This map has finite kernel. Assembling local computations, we have

eqn:hat-heqn:hat-h (6.14) HΣ(χ, s) =

∏kj=1 L(sj, χj)

QΣ(χ, s)ζΣ,∞(s, χ),

where QΣ(χ, s) bounded uniformly in χ, in compact subsets in <(sj) >1/2 + δ, δ > 0, and

|ζΣ,∞(s, χ)| 1

(1 + ‖m‖)d+1· 1

(1 + ‖χ‖)d′+1.

This implies that

ZΣ(s) =

∫MΓ

R

fΣ(s + im)dm,

wherefΣ(s) :=

∑χ∈(T 1(AF )/KT ·T (F ))∗

HΣ(χ, s)

We have

(1) (s1 − 1) · . . . · (sk − 1)fΣ(s) is homolorphic for <(sj) > 1− δ;(2) fΣ satisfies growth conditions in vertical strips (this follows by

applying the Phragmen-Lindelof principle 6.1.2 to bound L-functions appearing in equation (6.14));

(3) limsj→1 fΣ(s) = c(fΣ) 6= 0.

The Convexity principle 6.1.1 implies a meromorphic continuationof Z(s) to a tubular neighborhood of the shifted cone Λeff(XΣ). Theidentification of the remaining factors β and τ requires another appli-cation of the Poisson formula. Other line bundles require a version ofthe technical theorem above.

sect:unipotent6.6. Unipotent groups. Let X ⊃ G be an equivariant compactifi-cation of a unipotent group over a number field F and

X \G = D = ∪i∈IDi.

Throughout, we will assume that G acts on X on both sides, i.e., thatX is a compactification of G × G/G, or a bi-equvariant compactifica-tion. We also assume that D is a divisor with normal crossings andits components Di are geometrically irreducible. The main geometricinvariants of X have been computed in Example 1.1.4: The Picard

86 YURI TSCHINKEL

group is freely generated by the classes of Di, the effective cone issimplicial, and the anticanonical class is sum of boundary componentswith nonnegative coefficients.

Local and global heights have been defined in Example 4.10.6:

HDi,v(x) := ‖fi(x)‖−1v and HDi(x) =

∏v

HDi,v(x),

where fi is the unique G-invariant section of H0(X,Di). We get a heightpairing:

H : G(AF )× Pic(X)C → Cas in Section 6.3. The bi-equivariance of X implies that H is invariantunder the action on both sides of a compact open subgroup K of thefinite adeles. Moreover, we can arrange that Hv is smooth in gv forarchimedean v.

The height zeta function

Z(s, g) :=∑

γ∈G(F )

H(s, g)−1

is holomorphic in s, for <(s) 0. As a function of g it is continu-ous and in L2(G((F )\G(AF )), for these s. We proceed to analyze itsspectral decomposition. We get a formal identity

eqn:formalzeqn:formalz (6.15) Z(s; g) =∑%

Z%(s; g),

where the sum is over all irreducible unitary representations (%,H%)of G(AF ) occuring in the right regular representation of G(AF ) inL2(G(F )\G(AF )). They are parametrized by F -rational orbits O = O%under the coadjoint action of G on the dual of its Lie algebra g∗. Therelevant orbits are integral - there exists a lattice in g∗(F ) such thatZ%(s; g) = 0 unless the intersection of O with this lattice is nonempty.The pole of highest order is contributed by the trivial representationand integrality insures that this representation is “isolated”.

Let % be an integral representation as above. It has the followingexplicit realization: There exists an F -rational subgroup M ⊂ G suchthat

% = IndGM(ψ),

where ψ is a certain character of M(AF ). In particular, for the trivialrepresentation, M = G and ψ is the trivial character. Further, there


exists a finite set of valuations S = S% such that dim(%v) = 1 for v /∈ Sand consequently

eqn:zetaeqn:zeta (6.16) Z%(s; g′) = ZS(s; g′) · ZS(s; g′).

It turns out that

ZS(s; g′) :=∏v/∈S

∫M(Fv)

Hv(s; gvg′v)−1ψ(gv)dgv,

with an appropriately normalized Haar measure dgv on M(Fv). Thefunction ZS is the projection of Z to ⊗v∈S%v.

The first key result is the explicit computation of height integrals:∫M(Fv)

Hv(s; gvg′v)−1ψ(gv)dgv

for almost all v. This has been done in [?] for equivariant compact-ifications of additive groups Gn

a (see Section 6.4); the same approachworks here too. The contribution from the trivial representation canbe computed using the formula of Denef-Loeser, as in (6.10):∫

G(AF )

H(s; g)−1dg =∏i

ζF (si − κi + 1) · Φ(s),

where Φ(s) is holomorphic for <(s) ∈ T−KX−ε, for some ε > 0, and−KX =

∑i κiDi. As in the case of additive groups in Section 6.4, this

term gives the “correct” pole at −KX . The analysis of 1-dimensionalrepresentations, with M = G, is similar to the additive case. Newdifficulties arise from infinite-dimensional % on the right side of theexpansion (6.15).

Next we need to estimate dim(%v) and the local integrals for nonar-chimedean v ∈ S%. The key result here is that the contribition tothe Euler product from these places is a holomorphic function whichcan be bounded from above by a polynomial in the coordinates of %,for <(s) ∈ T−KX−ε. The uniform convergence of the spectral expan-sion comes from estimates at the archimedean places: for every (leftor right) G-invariant differential operator ∂ (and s ∈ T) there exists aconstant c(∂) such that

eqn:ceqn:c (6.17)

∫G(Fv)

|∂Hv(s; gv)−1dgv|v ≤ c(∂).

88 YURI TSCHINKEL

Let v be real. It is known that %v can be modeled in L2(Rr), where2r = dim(O). More precisely, there exists an isometry

j : (πv, L2(Rr))→ (%v,Hv)

(an analog of the Θ-distribution). Moreover, the universal envelopingalgebra U(g) surjects onto the Weyl algebra of differential operatorswith polynomial coefficients acting on the smooth vectors C∞(Rr) ⊂L2(Rr). In particular, we can find an operator ∆ acting as the (r-dimensional) harmonic oscillator

r∏j=1

(∂2

∂x2j

− ajx2j

),

with aj > 0. We choose an orthonormal basis of L2(Rr) consisting of∆-eigenfunctions ωλ (which are well known) and analyze∫

G(Fv)

Hv(s; gv)−1ωλ(gv)dgv,

where ωλ = j−1(ωλ). Using integration by parts we find that for s ∈ Tand any N ∈ N there is a constant c(N,∆) such that this integral isbounded by

eqn:boundeqn:bound (6.18) (1 + |λ|)−Nc(N,∆).

This estimate suffices to conclude that for each % the function ZS% isholomorphic in T.

Now the issue is to prove the convergence of the sum in (6.15). Usingany element ∂ ∈ U(g) acting in H% by a scalar λ(∂) 6= 0 (for example,any element in the center of U(g)) we can improve the bound (6.18) to

(1 + |λ|)−N1λ(∂)−N2c(N1, N2,∆, ∂)

(for any N1, N2 ∈ N). However, we have to insure the uniformity ofsuch estimates over the set of all %. This relies on a parametrizationof coadjoint orbits. There is a finite set Σd of “packets” of coadjointorbits, each parametrized by a locally closed subvariety Zd ⊂ g∗, andfor each d a finite set of F -rational polynomials Pd,r on g∗ such thatthe restriction of each Pd,r to Zd is invariant under the coadjoint action.Consequenty, the corresponding derivatives

∂d,r ∈ U(g)

act in H% by multiplication by the scalar

λ%,r = Pd,r(O).


There is a similar uniform construction of the “harmonic oscillator” ∆d

for each d. Combining the resulting estimates we obtain the uniformconvergence of the right hand side in (6.15).

The last technical point is to prove that both expressions (6.6) and(6.15) for Z(s; g) define continuous functions on G(F )\G(AF ). Then(6.15) gives the desired meromorphic continuation of Z(s; e).

Background material on representation theory of unipotent groupscan be found in the books [CG66], [Dix96] and the papers [Moo65],[Kir99].

sect:semisimple6.7. Semisimple groups.

sect:homogeneous6.8. Homogeneous spaces.

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Courant Institute, 251 Mercer Street, New York, NY 10012, USAand Mathematisches Institut, Georg-August-Universitat Gottingen,Bunsenstrasse 3-5, D-37073 Gottingen, Germany

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